%------------------------------------------------------------------------------
% File     : ITP229_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Insert 00617_038319
% 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    : 0066_VEBT_Insert_00617_038319 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11547 (4056 unt;1632 typ;   0 def)
%            Number of atoms       : 18790 (8095 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 20648 (2170   ~; 373   |;2166   &)
%                                         (2159 <=>;13780  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   14 (   2 avg)
%            Number of FOOLs       :  736 ( 437 fml; 299 var)
%            Number of X terms     :  584 (   0  []; 517 ite;  67 let)
%            Number of types       :   11 (  10 usr)
%            Number of type conns  : 1391 (1182   >; 209   *;   0   +;   0  <<)
%            Number of predicates  :  257 ( 254 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1381 (1381 usr;  63 con; 0-7 aty)
%            Number of variables   : 36191 (32600   !; 990   ?;36191   :)
%                                         (2601  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 20:04:04.244
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $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_Set_Oset,type,
    set: $tType > $tType ).

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

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

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

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

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

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

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

% Explicit typings (1614)
tff(sy_cl_HOL_Otype,type,
    type: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ps____,type,
    aTP_Lamp_ps: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pt____,type,
    aTP_Lamp_pt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pu____,type,
    aTP_Lamp_pu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pv____,type,
    aTP_Lamp_pv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pw____,type,
    aTP_Lamp_pw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__px____,type,
    aTP_Lamp_px: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__py____,type,
    aTP_Lamp_py: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pz____,type,
    aTP_Lamp_pz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qa____,type,
    aTP_Lamp_qa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qb____,type,
    aTP_Lamp_qb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qc____,type,
    aTP_Lamp_qc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qd____,type,
    aTP_Lamp_qd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qe____,type,
    aTP_Lamp_qe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qf____,type,
    aTP_Lamp_qf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qg____,type,
    aTP_Lamp_qg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qh____,type,
    aTP_Lamp_qh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qi____,type,
    aTP_Lamp_qi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qj____,type,
    aTP_Lamp_qj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xi____,type,
    aTP_Lamp_xi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xj____,type,
    aTP_Lamp_xj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xk____,type,
    aTP_Lamp_xk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xl____,type,
    aTP_Lamp_xl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xm____,type,
    aTP_Lamp_xm: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yg____,type,
    aTP_Lamp_yg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yh____,type,
    aTP_Lamp_yh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yv____,type,
    aTP_Lamp_yv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yw____,type,
    aTP_Lamp_yw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yy____,type,
    aTP_Lamp_yy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zd____,type,
    aTP_Lamp_zd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ze____,type,
    aTP_Lamp_ze: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : fun(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__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_Omax2,type,
    bNF_We1388413361240627857o_max2: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
    basic_BNF_size_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun(nat,nat) ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( A > fun(nat,A) ) ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,$o) ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
    boolea2506097494486148201lgebra: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
    boolea3799213064322606851m_diff: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).

tff(sy_c_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__class_Oiteratesp,type,
    comple7512665784863727008ratesp: 
      !>[A: $tType] : ( fun(A,A) > fun(A,$o) ) ).

tff(sy_c_Complete__Partial__Order_Ochain,type,
    comple1602240252501008431_chain: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Complex_OArg,type,
    arg: complex > real ).

tff(sy_c_Complex_Ocis,type,
    cis: real > complex ).

tff(sy_c_Complex_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_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: $tType] : ( ( fun(A,$o) * 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_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: $tType] : fun(set(A),$o) ).

tff(sy_c_Finite__Set_Ofold,type,
    finite_fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).

tff(sy_c_Finite__Set_Ofold__graph,type,
    finite_fold_graph: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > fun(B,$o) ) ).

tff(sy_c_Finite__Set_Ofolding__idem__on,type,
    finite1890593828518410140dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
    finite6916993218817215295axioms: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on,type,
    finite_folding_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on_OF,type,
    finite_folding_F: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B ) > fun(set(A),B) ) ).

tff(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(fun(A,B),fun(A,C)) ) ).

tff(sy_c_Fun_Ofun__upd,type,
    fun_upd: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).

tff(sy_c_Fun_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Omap__fun,type,
    map_fun: 
      !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(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_Ois__measure,type,
    fun_is_measure: 
      !>[A: $tType] : ( fun(A,nat) > $o ) ).

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_Fun__Def_Orp__inv__image,type,
    fun_rp_inv_image: 
      !>[A: $tType,B: $tType] : fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))) ).

tff(sy_c_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_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] : ( 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(B,A) * A * list(B) ) > A ) ).

tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : ( 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: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : 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_Onat,type,
    nat2: fun(int,nat) ).

tff(sy_c_Int_Opower__int,type,
    power_int: 
      !>[A: $tType] : ( ( A * int ) > A ) ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : 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: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > 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] : ( list(list(A)) > list(A) ) ).

tff(sy_c_List_Ocoset,type,
    coset: 
      !>[A: $tType] : ( list(A) > set(A) ) ).

tff(sy_c_List_Ocount__list,type,
    count_list: 
      !>[A: $tType] : ( list(A) > fun(A,nat) ) ).

tff(sy_c_List_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

tff(sy_c_List_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OdropWhile,type,
    dropWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),list(A)) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(A) ) ).

tff(sy_c_List_Ofold,type,
    fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * set(B) * fun(B,A) ) > $o ) ).

tff(sy_c_List_Ofoldl,type,
    foldl: 
      !>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

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: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > 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: $tType] : ( fun(A,fun(A,$o)) > fun(fun(B,A),fun(set(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(B,A) * 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_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > 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: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(list(A),list(Aa)) ) ).

tff(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : fun(list(A),set(A)) ).

tff(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( fun(A,nat) > 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] : ( ( list(A) * nat * A ) > list(A) ) ).

tff(sy_c_List_Olistrel,type,
    listrel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).

tff(sy_c_List_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olistrel1p,type,
    listrel1p: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

tff(sy_c_List_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_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Oord_Olexordp,type,
    lexordp2: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(list(A),fun(list(A),$o)) ) ).

tff(sy_c_List_Oord__class_Olexordp,type,
    ord_lexordp: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_Opartition,type,
    partition: 
      !>[A: $tType] : ( fun(A,$o) > 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] : ( 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: $tType] : fun(list(A),fun(list(A),$o)) ).

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] : ( ( nat * 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] : ( 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: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).

tff(sy_c_List_Osplice,type,
    splice: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Osplice__rel,type,
    splice_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otake,type,
    take: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),$o)) ).

tff(sy_c_List_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Nat_Ofunpow,type,
    funpow: 
      !>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,$o) ) ).

tff(sy_c_Nat_Osemiring__1__class_ONats,type,
    semiring_1_Nats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: fun(nat,int) ).

tff(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: fun(int,nat) ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: fun(list(nat),nat) ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun(real,real) ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun(real,real) ).

tff(sy_c_Num_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_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : 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_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: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),A) ) ).

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
    order_strict_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Partial__Function_Oflat__lub,type,
    partial_flat_lub: 
      !>[A: $tType] : ( ( A * set(A) ) > A ) ).

tff(sy_c_Power_Opower_Opower,type,
    power2: 
      !>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * nat ) > A ) ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : ( A > fun(nat,A) ) ).

tff(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : 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(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_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,$o) ) ).

tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_Product__Type_Oprod_Oswap,type,
    product_swap: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).

tff(sy_c_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Rat_OFract,type,
    fract: ( int * int ) > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
    field_char_0_Rats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,$o) ).

tff(sy_c_Real_OReal,type,
    real2: fun(nat,rat) > real ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun(nat,rat) > $o ).

tff(sy_c_Real_Opositive,type,
    positive2: fun(real,$o) ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun(real,fun(nat,rat)) ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun(nat,rat) > $o ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

tff(sy_c_Real__Vector__Spaces_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : ( real > A ) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( real > fun(A,A) ) ).

tff(sy_c_Relation_OField,type,
    field2: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

tff(sy_c_Relation_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_Oinv__image,type,
    inv_image: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_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_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun($o,A) ).

tff(sy_c_Series_Osuminf,type,
    suminf: 
      !>[A: $tType] : ( fun(nat,A) > A ) ).

tff(sy_c_Series_Osummable,type,
    summable: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( ( fun(nat,A) * A ) > $o ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,$o),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image2: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Oremove,type,
    remove: 
      !>[A: $tType] : 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_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_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_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_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortrancl,type,
    transitive_rtrancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Olex__prod,type,
    lex_prod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Wellfounded_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),fun(set(A),$o)) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Owf,type,
    wf: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Wfrec_Osame__fst,type,
    same_fst: 
      !>[A: $tType,B: $tType] : ( ( fun(A,$o) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Zorn_OChains,type,
    chains: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).

tff(sy_c_Zorn_Ochain__subset,type,
    chain_subset: 
      !>[A: $tType] : ( set(set(A)) > $o ) ).

tff(sy_c_Zorn_Ochains,type,
    chains2: 
      !>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).

tff(sy_c_Zorn_Opred__on_Ochain,type,
    pred_chain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) ) > fun(set(A),$o) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_fNot,type,
    fNot: fun($o,$o) ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( ( A * set(A) ) > $o ) ).

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list(vEBT_VEBT) ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_ya____,type,
    ya: nat ).

% Relevant facts (9162)
tff(fact_0_False,axiom,
    ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),mi),ma) ).

% False
tff(fact_1__092_060open_062x_A_060_Ami_092_060close_062,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),xa),mi) ).

% \<open>x < mi\<close>
tff(fact_2__C4_Ohyps_C_I7_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),mi),ma) ).

% "4.hyps"(7)
tff(fact_3__C4_Oprems_C_I3_J,axiom,
    aa(nat,$o,vEBT_vebt_member(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),xa)),ya) ).

% "4.prems"(3)
tff(fact_4_True,axiom,
    ( ( ya = xa )
    | ( ya = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),mi),ma) ) ) ).

% True
tff(fact_5__092_060open_062y_A_061_Amax_Ami_Ama_092_060close_062,axiom,
    ya = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),mi),ma) ).

% \<open>y = max mi ma\<close>
tff(fact_6_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_minNull(Ta)
     => ~ aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_7_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_8_option_Oinject,axiom,
    ! [A: $tType,X2: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X2) = aa(A,option(A),some(A),Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_9_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_10_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: 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),A3),B3) )
    <=> ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_11__C4_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) ) ).

% "4.hyps"(6)
tff(fact_12_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod(nat,nat)] :
      ~ ! [K: nat,M: nat] : Xb != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K),M) ).

% prod_decode_aux.cases
tff(fact_13__C4_Ohyps_C_I1_J,axiom,
    vEBT_invar_vebt(summary,m) ).

% "4.hyps"(1)
tff(fact_14_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Yb: product_prod(A,B)] :
      ~ ! [A4: A,B4: B] : Yb != 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_15_surj__pair,axiom,
    ! [A: $tType,B: $tType,P: product_prod(A,B)] :
    ? [X3: A,Y: B] : P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) ).

% surj_pair
tff(fact_16_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_17_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Ta)
     => ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_12) ) ).

% not_min_Null_member
tff(fact_18_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_19_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_20_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Ta),Xb)
      <=> member(nat,Xb,vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_21_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Dega)
     => ( ( Mia = Maa )
       => ( ! [X: vEBT_VEBT] :
              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
             => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) )
          & ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_22_prod__induct7,axiom,
    ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C2),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D2),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2)))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o,P2,Xb) ) ).

% prod_induct7
tff(fact_23_prod__induct6,axiom,
    ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C2),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D2),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o,P2,Xb) ) ).

% prod_induct6
tff(fact_24_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P2,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P2,Xb) ) ).

% prod_induct5
tff(fact_25_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,D))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A4: A,B4: B,C2: C,D2: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P2,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2))))
     => aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P2,Xb) ) ).

% prod_induct4
tff(fact_26_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,C)),$o),Xb: product_prod(A,product_prod(B,C))] :
      ( ! [A4: A,B4: B,C2: C] : aa(product_prod(A,product_prod(B,C)),$o,P2,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2)))
     => aa(product_prod(A,product_prod(B,C)),$o,P2,Xb) ) ).

% prod_induct3
tff(fact_27_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Yb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : Yb != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C2),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D2),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2)))))) ).

% prod_cases7
tff(fact_28_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Yb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : Yb != 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))),C2),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)),D2),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2))))) ).

% prod_cases6
tff(fact_29_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Yb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] : Yb != 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)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).

% prod_cases5
tff(fact_30_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Yb: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A4: A,B4: B,C2: C,D2: D] : Yb != 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),C2),D2))) ).

% prod_cases4
tff(fact_31_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Yb: product_prod(A,product_prod(B,C))] :
      ~ ! [A4: A,B4: B,C2: C] : Yb != 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),C2)) ).

% prod_cases3
tff(fact_32_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: 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),A3),B3) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_33_prod__cases,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),$o),P: product_prod(A,B)] :
      ( ! [A4: A,B4: B] : aa(product_prod(A,B),$o,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))
     => aa(product_prod(A,B),$o,P2,P) ) ).

% prod_cases
tff(fact_34__C4_Ohyps_C_I3_J,axiom,
    m = na ).

% "4.hyps"(3)
tff(fact_35_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb3
tff(fact_36_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_37_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_38_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_39_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_40_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% max.bounded_iff
tff(fact_41_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : ~ vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_42_old_Oprod_Orec,axiom,
    ! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A2: B,B2: C] : product_rec_prod(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F1,A2),B2) ).

% old.prod.rec
tff(fact_43_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P2: fun(A,$o)] :
      ( member(A,A2,aa(fun(A,$o),set(A),collect(A),P2))
    <=> aa(A,$o,P2,A2) ) ).

% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A5: set(A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_a(set(A),fun(A,$o),A5)) = A5 ).

% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
    ! [A: $tType,P2: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(A,$o,P2,X3)
        <=> aa(A,$o,Q,X3) )
     => ( aa(fun(A,$o),set(A),collect(A),P2) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_47_ext,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X3: A] : aa(A,B,F3,X3) = aa(A,B,G3,X3)
     => ( F3 = G3 ) ) ).

% ext
tff(fact_48_max_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A2),A2) = A2 ) ).

% max.idem
tff(fact_49_max_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ).

% max.left_idem
tff(fact_50_max_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ).

% max.right_idem
tff(fact_51__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_52_max_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)) ) ).

% max.left_commute
tff(fact_53_max_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A2) ) ).

% max.commute
tff(fact_54_max_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)) ) ).

% max.assoc
tff(fact_55_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_56_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_57_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_58_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_59_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Yb) ) ) ) ).

% le_max_iff_disj
tff(fact_60_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded2
tff(fact_61_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded1
tff(fact_62_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_63_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2) ) ) ) ).

% max.boundedI
tff(fact_64_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% max.boundedE
tff(fact_65_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% max.orderI
tff(fact_66_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.orderE
tff(fact_67_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_68_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_69_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_70_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_71_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% max.strict_boundedE
tff(fact_72_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Yb) ) ) ) ).

% less_max_iff_disj
tff(fact_73_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_VEBT_valid(Ta,D3)
     => vEBT_invar_vebt(Ta,D3) ) ).

% valid_eq2
tff(fact_74_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_invar_vebt(Ta,D3)
     => vEBT_VEBT_valid(Ta,D3) ) ).

% valid_eq1
tff(fact_75_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_VEBT_valid(Ta,D3)
    <=> vEBT_invar_vebt(Ta,D3) ) ).

% valid_eq
tff(fact_76_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% dual_order.refl
tff(fact_77_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Xb) ) ).

% order_refl
tff(fact_78_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(set(nat),$o,finite_finite2(nat),vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_79_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% deg_not_0
tff(fact_80_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb) = Yb ) ) ) ).

% max_absorb2
tff(fact_81_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb) = Xb ) ) ) ).

% max_absorb1
tff(fact_82_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),B2,A2) ) ).

% max_def
tff(fact_83_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P2: fun(A,$o),K2: A,F3: fun(A,nat),B2: nat] :
      ( aa(A,$o,P2,K2)
     => ( ! [Y: A] :
            ( aa(A,$o,P2,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Y)),B2) )
       => ? [X3: A] :
            ( aa(A,$o,P2,X3)
            & ! [Y3: A] :
                ( aa(A,$o,P2,Y3)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Y3)),aa(A,nat,F3,X3)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_84_nat__descend__induct,axiom,
    ! [Nb: nat,P2: fun(nat,$o),Mb: nat] :
      ( ! [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
         => aa(nat,$o,P2,K) )
     => ( ! [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( ! [I: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),I)
                 => aa(nat,$o,P2,I) )
             => aa(nat,$o,P2,K) ) )
       => aa(nat,$o,P2,Mb) ) ) ).

% nat_descend_induct
tff(fact_85_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_86_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_87_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P2: fun(A,$o),K2: A,F3: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P2,K2)
     => ( ! [X3: A] :
            ( aa(A,$o,P2,X3)
           => ? [Y3: A] :
                ( aa(A,$o,P2,Y3)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Y3)),aa(A,nat,F3,X3)) ) )
       => ? [Y: A] :
            ( aa(A,$o,P2,Y)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Y)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,K2)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_88_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts,S),Xb) = vEBT_Node(Info,zero_zero(nat),Ts,S) ).

% vebt_insert.simps(2)
tff(fact_89_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_90_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Yb) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Yb)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_91_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( ( Xb = Yb )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ) ).

% order_class.order_eq_iff
tff(fact_92_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% ord_eq_le_trans
tff(fact_93_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( B2 = C3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% ord_le_eq_trans
tff(fact_94_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
           => ( Xb = Yb ) ) ) ) ).

% order_antisym
tff(fact_95_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% order.trans
tff(fact_96_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) ) ) ) ).

% order_trans
tff(fact_97_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),P2,A4),B4) )
         => ( ! [A4: A,B4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P2,B4),A4)
               => aa(A,$o,aa(A,fun(A,$o),P2,A4),B4) )
           => aa(A,$o,aa(A,fun(A,$o),P2,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_98_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_99_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_100_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% dual_order.trans
tff(fact_101_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_102_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,G3,Xb)) ) ) ).

% le_funD
tff(fact_103_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,G3,Xb)) ) ) ).

% le_funE
tff(fact_104_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3) ) ) ).

% le_funI
tff(fact_105_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
        <=> ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4)) ) ) ).

% le_fun_def
tff(fact_106_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_107_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_subst1
tff(fact_108_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,B2)),C3)
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_subst2
tff(fact_109_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A] :
          ( ( Xb = Yb )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ).

% order_eq_refl
tff(fact_110_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ).

% linorder_linear
tff(fact_111_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A2 = aa(B,A,F3,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_112_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_113_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ).

% linorder_le_cases
tff(fact_114_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
          <=> ( Xb = Yb ) ) ) ) ).

% order_antisym_conv
tff(fact_115_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] :
        ? [Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ).

% lt_ex
tff(fact_116_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xb: A] :
        ? [X_12: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X_12) ) ).

% gt_ex
tff(fact_117_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ? [Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Yb) ) ) ) ).

% dense
tff(fact_118_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( Xb != Yb ) ) ) ).

% less_imp_neq
tff(fact_119_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order.asym
tff(fact_120_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% ord_eq_less_trans
tff(fact_121_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( B2 = C3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% ord_less_eq_trans
tff(fact_122_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X3)
                 => aa(A,$o,P2,Y3) )
             => aa(A,$o,P2,X3) )
         => aa(A,$o,P2,A2) ) ) ).

% less_induct
tff(fact_123_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Yb: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
          <=> ( Xb = Yb ) ) ) ) ).

% antisym_conv3
tff(fact_124_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( ( Xb != Yb )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ) ).

% linorder_cases
tff(fact_125_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% dual_order.asym
tff(fact_126_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% dual_order.irrefl
tff(fact_127_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P2,X_13)
        <=> ? [N: A] :
              ( aa(A,$o,P2,N)
              & ! [M2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),N)
                 => ~ aa(A,$o,P2,M2) ) ) ) ) ).

% exists_least_iff
tff(fact_128_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),P2,A4),B4) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P2,A4),A4)
           => ( ! [A4: A,B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P2,B4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P2,A4),B4) )
             => aa(A,$o,aa(A,fun(A,$o),P2,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_129_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans
tff(fact_130_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
            | ( Xb = Yb ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_131_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_132_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_133_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_134_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ( Xb != Yb )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ) ).

% linorder_neqE
tff(fact_135_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ).

% order_less_asym
tff(fact_136_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ( Xb != Yb )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ) ).

% linorder_neq_iff
tff(fact_137_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order_less_asym'
tff(fact_138_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_less_trans
tff(fact_139_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A2 = aa(B,A,F3,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_140_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_141_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Xb) ) ).

% order_less_irrefl
tff(fact_142_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_less_subst1
tff(fact_143_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,B2)),C3)
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_less_subst2
tff(fact_144_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ).

% order_less_not_sym
tff(fact_145_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A,P2: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
           => (P2) ) ) ) ).

% order_less_imp_triv
tff(fact_146_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
          | ( Xb = Yb )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ).

% linorder_less_linear
tff(fact_147_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( Xb != Yb ) ) ) ).

% order_less_imp_not_eq
tff(fact_148_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( Yb != Xb ) ) ) ).

% order_less_imp_not_eq2
tff(fact_149_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ).

% order_less_imp_not_less
tff(fact_150_ex__has__least__nat,axiom,
    ! [A: $tType,P2: fun(A,$o),K2: A,Mb: fun(A,nat)] :
      ( aa(A,$o,P2,K2)
     => ? [X3: A] :
          ( aa(A,$o,P2,X3)
          & ! [Y3: A] :
              ( aa(A,$o,P2,Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Mb,X3)),aa(A,nat,Mb,Y3)) ) ) ) ).

% ex_has_least_nat
tff(fact_151_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_152_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb) ) ) ).

% leD
tff(fact_153_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ).

% leI
tff(fact_154_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_155_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
          <=> ( Xb = Yb ) ) ) ) ).

% antisym_conv1
tff(fact_156_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
          <=> ( Xb = Yb ) ) ) ) ).

% antisym_conv2
tff(fact_157_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Yb: A] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),X3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z) ) ) ).

% dense_ge
tff(fact_158_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Yb: A,Z: A] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Yb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z) ) ) ).

% dense_le
tff(fact_159_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ) ).

% less_le_not_le
tff(fact_160_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Yb: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb) ) ) ).

% not_le_imp_less
tff(fact_161_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_162_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_163_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans1
tff(fact_164_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans2
tff(fact_165_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% order.strict_iff_not
tff(fact_166_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Xb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),W) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_167_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Yb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z) ) ) ) ).

% dense_le_bounded
tff(fact_168_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_169_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_170_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_171_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_172_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_173_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% order.strict_implies_order
tff(fact_174_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% dual_order.strict_implies_order
tff(fact_175_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
            | ( Xb = Yb ) ) ) ) ).

% order_le_less
tff(fact_176_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
            & ( Xb != Yb ) ) ) ) ).

% order_less_le
tff(fact_177_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ).

% linorder_not_le
tff(fact_178_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ).

% linorder_not_less
tff(fact_179_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ).

% order_less_imp_le
tff(fact_180_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( A2 != B2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_181_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_182_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_le_less_trans
tff(fact_183_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_less_le_trans
tff(fact_184_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_185_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,B2)),C3)
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_le_less_subst2
tff(fact_186_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_187_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,B2)),C3)
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_less_le_subst2
tff(fact_188_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ).

% linorder_le_less_linear
tff(fact_189_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
            | ( Xb = Yb ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_190_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% add_gr_0
tff(fact_191_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Nb),Nb) ) ).

% buildup_gives_valid
tff(fact_192_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_193_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_194_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_195_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_196_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_197_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_198_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_199_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_200_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_201_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_202_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C3: 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),C3),A2) )
        <=> ( B2 = C3 ) ) ) ).

% add_right_cancel
tff(fact_203_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C3: 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),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add_left_cancel
tff(fact_204_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),zero_zero(A))
        <=> ( Nb = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_205_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb)
        <=> ( Nb = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_206_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_207_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_208_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_209_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Yb: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Yb = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_210_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Yb = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_211_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_212_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_213_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_214_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_215_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_216_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_217_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_218_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_219_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_220_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% neq0_conv
tff(fact_221_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_222_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% le0
tff(fact_223_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A2) ).

% bot_nat_0.extremum
tff(fact_224_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),zero_zero(nat)) = Mb ).

% Nat.add_0_right
tff(fact_225_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) = zero_zero(nat) )
    <=> ( ( Mb = zero_zero(nat) )
        & ( Nb = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_226_nat__add__left__cancel__less,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_227_nat__add__left__cancel__le,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_228_max__0R,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),zero_zero(nat)) = Nb ).

% max_0R
tff(fact_229_max__0L,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),Nb) = Nb ).

% max_0L
tff(fact_230_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_231_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_232_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_233_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_234_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_235_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_236_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F3),G3)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G3),F3) ) ) ) ).

% less_fun_def
tff(fact_237_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: A] :
          ( ( zero_zero(A) = Xb )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_238_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P2: fun(A,$o),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Y3)),aa(A,B,F3,X3))
                 => aa(A,$o,P2,Y3) )
             => aa(A,$o,P2,X3) )
         => aa(A,$o,P2,A2) ) ) ).

% measure_induct_rule
tff(fact_239_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P2: fun(A,$o),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Y3)),aa(A,B,F3,X3))
                 => aa(A,$o,P2,Y3) )
             => aa(A,$o,P2,X3) )
         => aa(A,$o,P2,A2) ) ) ).

% measure_induct
tff(fact_240_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C3: 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),C3),A2) )
         => ( B2 = C3 ) ) ) ).

% add_right_imp_eq
tff(fact_241_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C3: 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),C3) )
         => ( B2 = C3 ) ) ) ).

% add_left_imp_eq
tff(fact_242_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.left_commute
tff(fact_243_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_244_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C3: 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),C3),A2) )
        <=> ( B2 = C3 ) ) ) ).

% add.right_cancel
tff(fact_245_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: 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),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add.left_cancel
tff(fact_246_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.assoc
tff(fact_247_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: A,K2: A,B2: A,A2: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_248_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: A,K2: A,A2: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_249_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( ( I2 = J )
            & ( K2 = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_250_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_251_infinite__descent__measure,axiom,
    ! [A: $tType,P2: fun(A,$o),V2: fun(A,nat),Xb: A] :
      ( ! [X3: A] :
          ( ~ aa(A,$o,P2,X3)
         => ? [Y3: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y3)),aa(A,nat,V2,X3))
              & ~ aa(A,$o,P2,Y3) ) )
     => aa(A,$o,P2,Xb) ) ).

% infinite_descent_measure
tff(fact_252_linorder__neqE__nat,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb != Yb )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Yb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Yb),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_253_infinite__descent,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( ! [N2: nat] :
          ( ~ aa(nat,$o,P2,N2)
         => ? [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
              & ~ aa(nat,$o,P2,M3) ) )
     => aa(nat,$o,P2,Nb) ) ).

% infinite_descent
tff(fact_254_nat__less__induct,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
             => aa(nat,$o,P2,M3) )
         => aa(nat,$o,P2,N2) )
     => aa(nat,$o,P2,Nb) ) ).

% nat_less_induct
tff(fact_255_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_irrefl_nat
tff(fact_256_less__not__refl3,axiom,
    ! [S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S),Ta)
     => ( S != Ta ) ) ).

% less_not_refl3
tff(fact_257_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_258_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_not_refl
tff(fact_259_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_260_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: fun(nat,$o),K2: nat,B2: nat] :
      ( aa(nat,$o,P2,K2)
     => ( ! [Y: nat] :
            ( aa(nat,$o,P2,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),B2) )
       => ? [X3: nat] :
            ( aa(nat,$o,P2,X3)
            & ! [Y3: nat] :
                ( aa(nat,$o,P2,Y3)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_261_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ) ).

% nat_le_linear
tff(fact_262_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_263_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% eq_imp_le
tff(fact_264_le__trans,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),K2) ) ) ).

% le_trans
tff(fact_265_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Nb) ).

% le_refl
tff(fact_266_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) ).

% zero_le
tff(fact_267_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb)
        <=> ( Nb != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_268_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Mb: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb)
         => ( Nb != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_269_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Nb),zero_zero(A)) ) ).

% not_less_zero
tff(fact_270_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ( Nb != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb) ) ) ).

% gr_zeroI
tff(fact_271_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_272_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_273_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ? [C4: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ).

% le_iff_add
tff(fact_274_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ) ).

% add_right_mono
tff(fact_275_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ~ ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ) ).

% less_eqE
tff(fact_276_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% add_left_mono
tff(fact_277_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_mono
tff(fact_278_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_279_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( ( I2 = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_280_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J)
            & ( K2 = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_281_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_282_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_283_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_284_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_285_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_286_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ) ).

% add_strict_right_mono
tff(fact_287_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% add_strict_left_mono
tff(fact_288_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_strict_mono
tff(fact_289_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J)
            & ( K2 = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_290_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( ( I2 = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_291_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_292_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P2: fun(A,$o),Xb: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V2,X3) = zero_zero(nat) )
         => aa(A,$o,P2,X3) )
     => ( ! [X3: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V2,X3))
           => ( ~ aa(A,$o,P2,X3)
             => ? [Y3: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y3)),aa(A,nat,V2,X3))
                  & ~ aa(A,$o,P2,Y3) ) ) )
       => aa(A,$o,P2,Xb) ) ) ).

% infinite_descent0_measure
tff(fact_293_infinite__descent0,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( ~ aa(nat,$o,P2,N2)
             => ? [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
                  & ~ aa(nat,$o,P2,M3) ) ) )
       => aa(nat,$o,P2,Nb) ) ) ).

% infinite_descent0
tff(fact_294_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( Nb != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_295_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_296_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% not_less0
tff(fact_297_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_298_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% gr0I
tff(fact_299_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_300_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),zero_zero(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_301_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_302_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_303_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% less_eq_nat.simps(1)
tff(fact_304_less__mono__imp__le__mono,axiom,
    ! [F3: fun(nat,nat),I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F3,I3)),aa(nat,nat,F3,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F3,I2)),aa(nat,nat,F3,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_305_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( ( Mb != Nb )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_306_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Mb = Nb ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_307_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_308_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_309_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_310_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) = Mb )
     => ( Nb = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_311_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),Nb) = Nb ).

% plus_nat.add_0
tff(fact_312_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_313_trans__less__add2,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),J)) ) ).

% trans_less_add2
tff(fact_314_trans__less__add1,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Mb)) ) ).

% trans_less_add1
tff(fact_315_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) ) ).

% add_less_mono1
tff(fact_316_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),I2) ).

% not_add_less2
tff(fact_317_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),I2) ).

% not_add_less1
tff(fact_318_add__less__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_319_add__lessD1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K2) ) ).

% add_lessD1
tff(fact_320_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
    <=> ? [K3: nat] : Nb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K3) ) ).

% nat_le_iff_add
tff(fact_321_trans__le__add2,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),J)) ) ).

% trans_le_add2
tff(fact_322_trans__le__add1,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Mb)) ) ).

% trans_le_add1
tff(fact_323_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) ) ).

% add_le_mono1
tff(fact_324_add__le__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_325_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),L)
     => ? [N2: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N2) ) ).

% le_Suc_ex
tff(fact_326_add__leD2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb) ) ).

% add_leD2
tff(fact_327_add__leD1,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% add_leD1
tff(fact_328_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% le_add2
tff(fact_329_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)) ).

% le_add1
tff(fact_330_add__leE,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)),Nb)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb) ) ) ).

% add_leE
tff(fact_331_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),ord_max(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ).

% max_add_distrib_right
tff(fact_332_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Yb),Z)) ) ).

% max_add_distrib_left
tff(fact_333_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q2)) ).

% nat_add_max_right
tff(fact_334_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Q2)) ).

% nat_add_max_left
tff(fact_335_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Yb = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_336_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Yb = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_337_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_338_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_339_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_increasing2
tff(fact_340_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ) ) ).

% add_decreasing2
tff(fact_341_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_increasing
tff(fact_342_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ) ) ).

% add_decreasing
tff(fact_343_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_less_le_mono
tff(fact_344_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_le_less_mono
tff(fact_345_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_346_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_347_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% pos_add_strict
tff(fact_348_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ ! [C2: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
               => ( C2 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_349_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_pos
tff(fact_350_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_351_ex__least__nat__le,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P2,Nb)
     => ( ~ aa(nat,$o,P2,zero_zero(nat))
       => ? [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
            & ! [I: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
               => ~ aa(nat,$o,P2,I) )
            & aa(nat,$o,P2,K) ) ) ) ).

% ex_least_nat_le
tff(fact_352_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ? [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_353_mono__nat__linear__lb,axiom,
    ! [F3: fun(nat,nat),Mb: nat,K2: nat] :
      ( ! [M: nat,N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F3,M)),aa(nat,nat,F3,N2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F3,Mb)),K2)),aa(nat,nat,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2))) ) ).

% mono_nat_linear_lb
tff(fact_354_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_strict_increasing2
tff(fact_355_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_strict_increasing
tff(fact_356_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_nonneg
tff(fact_357_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_358_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_pos
tff(fact_359_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_360_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),$o,finite_finite2(A),aa(list(A),set(A),set2(A),Xs)) ).

% List.finite_set
tff(fact_361_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_362_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_leaf
tff(fact_363_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Yb),E2)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ).

% field_le_epsilon
tff(fact_364_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_365_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_366_finite__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A5: set(A)] : aa(set(A),$o,finite_finite2(A),A5) ) ).

% finite_code
tff(fact_367_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_min_max
tff(fact_368_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N3)
    <=> ? [M2: nat] :
        ! [X4: nat] :
          ( member(nat,X4,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),M2) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_369_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ! [M2: nat] :
        ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
          & member(nat,N,S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_370_finite__nat__set__iff__bounded,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N3)
    <=> ? [M2: nat] :
        ! [X4: nat] :
          ( member(nat,X4,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),M2) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_371_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( vEBT_V5719532721284313246member(Ta,Xb)
        | vEBT_VEBT_membermima(Ta,Xb) ) ) ).

% both_member_options_def
tff(fact_372_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [Xb: B] : aa(B,A,bot_bot(fun(B,A)),Xb) = bot_bot(A) ) ).

% bot_apply
tff(fact_373_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tree,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
       => ( vEBT_V5719532721284313246member(Tree,Xb)
          | vEBT_VEBT_membermima(Tree,Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_374_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),Xb) = Xb ) ).

% max_bot
tff(fact_375_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),bot_bot(A)) = Xb ) ).

% max_bot2
tff(fact_376_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X: A] : aa(A,B,bot_bot(fun(A,B)),X) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_377_finite_OemptyI,axiom,
    ! [A: $tType] : aa(set(A),$o,finite_finite2(A),bot_bot(set(A))) ).

% finite.emptyI
tff(fact_378_infinite__imp__nonempty,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ( S2 != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_379_finite__transitivity__chain,axiom,
    ! [A: $tType,A5: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X3: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,X3),X3)
       => ( ! [X3: A,Y: A,Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R,X3),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),R,Y),Z2)
               => aa(A,$o,aa(A,fun(A,$o),R,X3),Z2) ) )
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => ? [Y3: A] :
                    ( member(A,Y3,A5)
                    & aa(A,$o,aa(A,fun(A,$o),R,X3),Y3) ) )
           => ( A5 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_380_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A2) ) ).

% bot.extremum
tff(fact_381_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_382_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_383_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_384_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A2) ) ) ).

% bot.not_eq_extremum
tff(fact_385_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,A5)
                & ! [Xa: A] :
                    ( member(A,Xa,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa)
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_386_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,A5)
                & ! [Xa: A] :
                    ( member(A,Xa,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X3)
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_387_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X5: set(A)] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => ? [Xa: A] :
                    ( member(A,Xa,X5)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xa) ) )
           => ~ aa(set(A),$o,finite_finite2(A),X5) ) ) ) ).

% infinite_growing
tff(fact_388_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,S2)
                & ~ ? [Xa: A] :
                      ( member(A,Xa,S2)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X3) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_389_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_390_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
        ? [Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ).

% linordered_field_no_lb
tff(fact_391_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
        ? [X_12: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X_12) ) ).

% linordered_field_no_ub
tff(fact_392_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( ( Xb != Yb )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_393_finite__subset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% finite_subset
tff(fact_394_infinite__super,axiom,
    ! [A: $tType,S2: set(A),T2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
     => ( ~ aa(set(A),$o,finite_finite2(A),S2)
       => ~ aa(set(A),$o,finite_finite2(A),T2) ) ) ).

% infinite_super
tff(fact_395_rev__finite__subset,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% rev_finite_subset
tff(fact_396_finite__psubset__induct,axiom,
    ! [A: $tType,A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [A6: set(A)] :
            ( aa(set(A),$o,finite_finite2(A),A6)
           => ( ! [B6: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B6),A6)
                 => aa(set(A),$o,P2,B6) )
             => aa(set(A),$o,P2,A6) ) )
       => aa(set(A),$o,P2,A5) ) ) ).

% finite_psubset_induct
tff(fact_397_finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A5: set(A)] : aa(set(A),$o,finite_finite2(A),A5) ) ).

% finite
tff(fact_398_finite__set__choice,axiom,
    ! [B: $tType,A: $tType,A5: set(A),P2: fun(A,fun(B,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X3: A] :
            ( member(A,X3,A5)
           => ? [X_1: B] : aa(B,$o,aa(A,fun(B,$o),P2,X3),X_1) )
       => ? [F2: fun(A,B)] :
          ! [X: A] :
            ( member(A,X,A5)
           => aa(B,$o,aa(A,fun(B,$o),P2,X),aa(A,B,F2,X)) ) ) ) ).

% finite_set_choice
tff(fact_399_bounded__Max__nat,axiom,
    ! [P2: fun(nat,$o),Xb: nat,M4: nat] :
      ( aa(nat,$o,P2,Xb)
     => ( ! [X3: nat] :
            ( aa(nat,$o,P2,X3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M4) )
       => ~ ! [M: nat] :
              ( aa(nat,$o,P2,M)
             => ~ ! [X: nat] :
                    ( aa(nat,$o,P2,X)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_400_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B5)
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => member(A,X4,B5) ) ) ).

% subset_code(1)
tff(fact_401_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F2: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc: A] : Xb != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),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)),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_402_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va: list(vEBT_VEBT),Vb: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),zero_zero(nat),Va,Vb),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_403_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ? [X3: A] :
                ( member(A,X3,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3)
                & ! [Xa: A] :
                    ( member(A,Xa,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa)
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_404_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ? [X3: A] :
                ( member(A,X3,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A2)
                & ! [Xa: A] :
                    ( member(A,Xa,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X3)
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_405_finite__list,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A5 ) ).

% finite_list
tff(fact_406_unbounded__k__infinite,axiom,
    ! [K2: nat,S2: set(nat)] :
      ( ! [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),M)
         => ? [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N4)
              & member(nat,N4,S2) ) )
     => ~ aa(set(nat),$o,finite_finite2(nat),S2) ) ).

% unbounded_k_infinite
tff(fact_407_bounded__nat__set__is__finite,axiom,
    ! [N3: set(nat),Nb: nat] :
      ( ! [X3: nat] :
          ( member(nat,X3,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),Nb) )
     => aa(set(nat),$o,finite_finite2(nat),N3) ) ).

% bounded_nat_set_is_finite
tff(fact_408_infinite__nat__iff__unbounded,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ! [M2: nat] :
        ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
          & member(nat,N,S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_409_empty__subsetI,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A5) ).

% empty_subsetI
tff(fact_410_subset__empty,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),bot_bot(set(A)))
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_411_empty__iff,axiom,
    ! [A: $tType,C3: A] : ~ member(A,C3,bot_bot(set(A))) ).

% empty_iff
tff(fact_412_all__not__in__conv,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [X4: A] : ~ member(A,X4,A5)
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_413_Collect__empty__eq,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P2) = bot_bot(set(A)) )
    <=> ! [X4: A] : ~ aa(A,$o,P2,X4) ) ).

% Collect_empty_eq
tff(fact_414_empty__Collect__eq,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),P2) )
    <=> ! [X4: A] : ~ aa(A,$o,P2,X4) ) ).

% empty_Collect_eq
tff(fact_415_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ~ ? [X: A] :
                  ( member(A,X,S2)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S2))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_416_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),Yb: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( member(A,Yb,S2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S2))),aa(A,B,F3,Yb)) ) ) ) ) ).

% arg_min_least
tff(fact_417_Euclid__induct,axiom,
    ! [P2: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P2,A4),B4)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P2,B4),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,A4),zero_zero(nat))
       => ( ! [A4: nat,B4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P2,A4),B4)
             => aa(nat,$o,aa(nat,fun(nat,$o),P2,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B4)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P2,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_418_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_419_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_420_psubsetI,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( ( A5 != B5 )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5) ) ) ).

% psubsetI
tff(fact_421_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
        | ( A5 = B5 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_422_subset__psubset__trans,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B5),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),C5) ) ) ).

% subset_psubset_trans
tff(fact_423_subset__not__subset__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5) ) ) ).

% subset_not_subset_eq
tff(fact_424_psubset__subset__trans,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),C5) ) ) ).

% psubset_subset_trans
tff(fact_425_psubset__imp__subset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% psubset_imp_subset
tff(fact_426_psubset__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
        & ( A5 != B5 ) ) ) ).

% psubset_eq
tff(fact_427_psubsetE,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5) ) ) ).

% psubsetE
tff(fact_428_psubsetD,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ( member(A,C3,A5)
       => member(A,C3,B5) ) ) ).

% psubsetD
tff(fact_429_psubset__trans,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B5),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),C5) ) ) ).

% psubset_trans
tff(fact_430_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_431_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_432_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% verit_la_disequality
tff(fact_433_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% verit_comp_simplify1(2)
tff(fact_434_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% verit_comp_simplify1(1)
tff(fact_435_ex__in__conv,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ? [X4: A] : member(A,X4,A5)
    <=> ( A5 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_436_equals0I,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [Y: A] : ~ member(A,Y,A5)
     => ( A5 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_437_equals0D,axiom,
    ! [A: $tType,A5: set(A),A2: A] :
      ( ( A5 = bot_bot(set(A)) )
     => ~ member(A,A2,A5) ) ).

% equals0D
tff(fact_438_emptyE,axiom,
    ! [A: $tType,A2: A] : ~ member(A,A2,bot_bot(set(A))) ).

% emptyE
tff(fact_439_not__psubset__empty,axiom,
    ! [A: $tType,A5: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_440_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => member(A,lattic7623131987881927897min_on(A,B,F3,S2),S2) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_441_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% verit_comp_simplify1(3)
tff(fact_442_subset__emptyI,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [X3: A] : ~ member(A,X3,A5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_443_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D22)
           => ? [E2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E2),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E2),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_444_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_445_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,P2,A2)
           => ( ~ aa(A,$o,P2,B2)
             => ? [C2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
                  & ! [X: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),C2) )
                     => aa(A,$o,P2,X) )
                  & ! [D4: A] :
                      ( ! [X3: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D4) )
                         => aa(A,$o,P2,X3) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D4),C2) ) ) ) ) ) ) ).

% complete_interval
tff(fact_446_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Ta) ) ) ).

% pinf(6)
tff(fact_447_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X) ) ) ).

% pinf(8)
tff(fact_448_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Ta) ) ) ).

% minf(6)
tff(fact_449_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X) ) ) ).

% minf(8)
tff(fact_450_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_451_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList: list(vEBT_VEBT),S3: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Nb)),TreeList,S3) ) ).

% deg_SUcn_Node
tff(fact_452_count__notin,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,nat,count_list(A,Xs),Xb) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_453_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N2: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2)
     => ~ ! [N2: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) ) ).

% even_odd_cases
tff(fact_454_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_455_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_456_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_457_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb)) ) ).

% Suc_mono
tff(fact_458_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_less_eq
tff(fact_459_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,suc,Mb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ) ).

% Suc_le_mono
tff(fact_460_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% add_Suc_right
tff(fact_461_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)) ).

% max_Suc_Suc
tff(fact_462_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_463_zero__less__Suc,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,Nb)) ).

% zero_less_Suc
tff(fact_464_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa(nat,nat,suc,Nb) ).

% n_not_Suc_n
tff(fact_465_Suc__inject,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa(nat,nat,suc,Xb) = aa(nat,nat,suc,Yb) )
     => ( Xb = Yb ) ) ).

% Suc_inject
tff(fact_466_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ( ( Xb != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va2: nat] : Xb != aa(nat,nat,suc,aa(nat,nat,suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_467_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ~ ! [N2: nat] : Xb != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_468_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_469_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_470_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_471_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_472_old_Onat_Oexhaust,axiom,
    ! [Yb: nat] :
      ( ( Yb != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Yb != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_473_nat__induct,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,P2,N2)
           => aa(nat,$o,P2,aa(nat,nat,suc,N2)) )
       => aa(nat,$o,P2,Nb) ) ) ).

% nat_induct
tff(fact_474_diff__induct,axiom,
    ! [P2: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [X3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,X3),zero_zero(nat))
     => ( ! [Y: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,zero_zero(nat)),aa(nat,nat,suc,Y))
       => ( ! [X3: nat,Y: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P2,X3),Y)
             => aa(nat,$o,aa(nat,fun(nat,$o),P2,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P2,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_475_zero__induct,axiom,
    ! [P2: fun(nat,$o),K2: nat] :
      ( aa(nat,$o,P2,K2)
     => ( ! [N2: nat] :
            ( aa(nat,$o,P2,aa(nat,nat,suc,N2))
           => aa(nat,$o,P2,N2) )
       => aa(nat,$o,P2,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_476_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa(nat,nat,suc,Mb) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_477_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_neq_Suc
tff(fact_478_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_not_Suc
tff(fact_479_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M: nat] : Nb = aa(nat,nat,suc,M) ) ).

% not0_implies_Suc
tff(fact_480_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K2)
     => ( ( K2 != aa(nat,nat,suc,I2) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( K2 != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_481_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_lessD
tff(fact_482_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),K2)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
           => ( K2 != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_483_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( ( aa(nat,nat,suc,Mb) != Nb )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_484_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_485_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb)) ) ).

% less_SucI
tff(fact_486_Ex__less__Suc,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P2,I4) )
    <=> ( aa(nat,$o,P2,Nb)
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
            & aa(nat,$o,P2,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_487_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_488_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb)) ) ).

% not_less_eq
tff(fact_489_All__less__Suc,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P2,I4) )
    <=> ( aa(nat,$o,P2,Nb)
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
           => aa(nat,$o,P2,I4) ) ) ) ).

% All_less_Suc
tff(fact_490_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Mb)
    <=> ? [M5: nat] :
          ( ( Mb = aa(nat,nat,suc,M5) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M5) ) ) ).

% Suc_less_eq2
tff(fact_491_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_492_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_493_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),K2) ) ) ).

% less_trans_Suc
tff(fact_494_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P2: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( ! [I3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,I3),aa(nat,nat,suc,I3))
       => ( ! [I3: nat,J2: nat,K: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P2,I3),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P2,J2),K)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P2,I3),K) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P2,I2),J) ) ) ) ).

% less_Suc_induct
tff(fact_495_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( ! [I3: nat] :
            ( ( J = aa(nat,nat,suc,I3) )
           => aa(nat,$o,P2,I3) )
       => ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J)
             => ( aa(nat,$o,P2,aa(nat,nat,suc,I3))
               => aa(nat,$o,P2,I3) ) )
         => aa(nat,$o,P2,I2) ) ) ) ).

% strict_inc_induct
tff(fact_496_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_497_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% Suc_leD
tff(fact_498_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ( Mb = aa(nat,nat,suc,Nb) ) ) ) ).

% le_SucE
tff(fact_499_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb)) ) ).

% le_SucI
tff(fact_500_Suc__le__D,axiom,
    ! [Nb: nat,M6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M6)
     => ? [M: nat] : M6 = aa(nat,nat,suc,M) ) ).

% Suc_le_D
tff(fact_501_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        | ( Mb = aa(nat,nat,suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_502_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_503_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_504_full__nat__induct,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M3)),N2)
             => aa(nat,$o,P2,M3) )
         => aa(nat,$o,P2,N2) )
     => aa(nat,$o,P2,Nb) ) ).

% full_nat_induct
tff(fact_505_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,P2,Mb)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),N2)
             => ( aa(nat,$o,P2,N2)
               => aa(nat,$o,P2,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,P2,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_506_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( ! [X3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X3),X3)
       => ( ! [X3: nat,Y: nat,Z2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X3),Y)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y),Z2)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X3),Z2) ) )
         => ( ! [N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N2),aa(nat,nat,suc,N2))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_507_nat__arith_Osuc1,axiom,
    ! [A5: nat,K2: nat,A2: nat] :
      ( ( A5 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),A2) )
     => ( aa(nat,nat,suc,A5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_508_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% add_Suc
tff(fact_509_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,suc,Nb)) ).

% add_Suc_shift
tff(fact_510_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Nb: nat,N5: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N2)),aa(nat,A,F3,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,N5)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_511_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Nb: nat,Mb: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N2)),aa(nat,A,F3,aa(nat,nat,suc,N2)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,Mb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_512_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Nb: nat,N5: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,F3,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,N5)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_513_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Nb: nat,N5: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N2))),aa(nat,A,F3,N2))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N5)),aa(nat,A,F3,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_514_Ex__less__Suc2,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P2,I4) )
    <=> ( aa(nat,$o,P2,zero_zero(nat))
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
            & aa(nat,$o,P2,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_515_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% gr0_conv_Suc
tff(fact_516_All__less__Suc2,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P2,I4) )
    <=> ( aa(nat,$o,P2,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
           => aa(nat,$o,P2,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_517_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ? [M: nat] : Nb = aa(nat,nat,suc,M) ) ).

% gr0_implies_Suc
tff(fact_518_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb))
    <=> ( ( Mb = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( Mb = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_519_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_520_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_le_eq
tff(fact_521_dec__induct,axiom,
    ! [I2: nat,J: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,P2,I2)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),J)
               => ( aa(nat,$o,P2,N2)
                 => aa(nat,$o,P2,aa(nat,nat,suc,N2)) ) ) )
         => aa(nat,$o,P2,J) ) ) ) ).

% dec_induct
tff(fact_522_inc__induct,axiom,
    ! [I2: nat,J: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,P2,J)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),J)
               => ( aa(nat,$o,P2,aa(nat,nat,suc,N2))
                 => aa(nat,$o,P2,N2) ) ) )
         => aa(nat,$o,P2,I2) ) ) ) ).

% inc_induct
tff(fact_523_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_524_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_525_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_526_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_527_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_528_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( Mb = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Nb = zero_zero(nat) ) )
        | ( ( Mb = zero_zero(nat) )
          & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_529_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) )
    <=> ( ( ( Mb = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Nb = zero_zero(nat) ) )
        | ( ( Mb = zero_zero(nat) )
          & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_530_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ~ ! [Q3: nat] : Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q3)) ) ).

% less_natE
tff(fact_531_less__add__Suc1,axiom,
    ! [I2: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Mb))) ).

% less_add_Suc1
tff(fact_532_less__add__Suc2,axiom,
    ! [I2: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I2))) ).

% less_add_Suc2
tff(fact_533_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
    <=> ? [K3: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_534_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ? [K: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)) ) ).

% less_imp_Suc_add
tff(fact_535_ex__least__nat__less,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P2,Nb)
     => ( ~ aa(nat,$o,P2,zero_zero(nat))
       => ? [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),Nb)
            & ! [I: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K)
               => ~ aa(nat,$o,P2,I) )
            & aa(nat,$o,P2,aa(nat,nat,suc,K)) ) ) ) ).

% ex_least_nat_less
tff(fact_536_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_537_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X) ) ) ).

% minf(7)
tff(fact_538_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Ta) ) ) ).

% minf(5)
tff(fact_539_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ( X != Ta ) ) ) ).

% minf(4)
tff(fact_540_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ( X != Ta ) ) ) ).

% minf(3)
tff(fact_541_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q4: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
             => ( aa(A,$o,P2,X3)
              <=> aa(A,$o,P3,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q4,X3) ) )
           => ? [Z2: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
               => ( ( aa(A,$o,P2,X)
                    | aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P3,X)
                    | aa(A,$o,Q4,X) ) ) ) ) ) ) ).

% minf(2)
tff(fact_542_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q4: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
             => ( aa(A,$o,P2,X3)
              <=> aa(A,$o,P3,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q4,X3) ) )
           => ? [Z2: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
               => ( ( aa(A,$o,P2,X)
                    & aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P3,X)
                    & aa(A,$o,Q4,X) ) ) ) ) ) ) ).

% minf(1)
tff(fact_543_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_544_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X) ) ) ).

% pinf(7)
tff(fact_545_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Ta) ) ) ).

% pinf(5)
tff(fact_546_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( X != Ta ) ) ) ).

% pinf(4)
tff(fact_547_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( X != Ta ) ) ) ).

% pinf(3)
tff(fact_548_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q4: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
             => ( aa(A,$o,P2,X3)
              <=> aa(A,$o,P3,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q4,X3) ) )
           => ? [Z2: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
               => ( ( aa(A,$o,P2,X)
                    | aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P3,X)
                    | aa(A,$o,Q4,X) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_549_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q4: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
             => ( aa(A,$o,P2,X3)
              <=> aa(A,$o,P3,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q4,X3) ) )
           => ? [Z2: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
               => ( ( aa(A,$o,P2,X)
                    & aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P3,X)
                    & aa(A,$o,Q4,X) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_550_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A2: A] :
        ? [B4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B4)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A2) ) ) ).

% ex_gt_or_lt
tff(fact_551_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% is_num_normalize(1)
tff(fact_552_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R2: A,S: B,R: set(product_prod(A,B)),S4: B] :
      ( 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)
     => ( ( S4 = S )
       => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_553_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xb) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_554_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_555_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X2: A] : size_option(A,Xb,aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_556_bot__empty__eq,axiom,
    ! [A: $tType,X: A] :
      ( aa(A,$o,bot_bot(fun(A,$o)),X)
    <=> member(A,X,bot_bot(set(A))) ) ).

% bot_empty_eq
tff(fact_557_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P2) = bot_bot(set(A)) )
    <=> ( P2 = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_558_option_Osize_I4_J,axiom,
    ! [A: $tType,X2: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_559_triangle__Suc,axiom,
    ! [Nb: nat] : nat_triangle(aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(Nb)),aa(nat,nat,suc,Nb)) ).

% triangle_Suc
tff(fact_560_dependent__nat__choice,axiom,
    ! [A: $tType,P2: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P2,zero_zero(nat)),X_1)
     => ( ! [X3: A,N2: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P2,N2),X3)
           => ? [Y3: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P2,aa(nat,nat,suc,N2)),Y3)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N2),X3),Y3) ) )
       => ? [F2: fun(nat,A)] :
          ! [N4: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P2,N4),aa(nat,A,F2,N4))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N4),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4))) ) ) ) ).

% dependent_nat_choice
tff(fact_561_exists__least__lemma,axiom,
    ! [P2: fun(nat,$o)] :
      ( ~ aa(nat,$o,P2,zero_zero(nat))
     => ( ? [X_1: nat] : aa(nat,$o,P2,X_1)
       => ? [N2: nat] :
            ( ~ aa(nat,$o,P2,N2)
            & aa(nat,$o,P2,aa(nat,nat,suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_562__092_060open_062vebt__insert_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Ix_M_Amax_Ami_Ama_J_J_Adeg_A_ItreeList_091high_Ami_An_A_058_061_Avebt__insert_A_ItreeList_A_B_Ahigh_Ami_An_J_A_Ilow_Ami_An_J_093_J_A_Iif_AminNull_A_ItreeList_A_B_Ahigh_Ami_An_J_Athen_Avebt__insert_Asummary_A_Ihigh_Ami_An_J_Aelse_Asummary_J_092_060close_062,axiom,
    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),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),xa),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),mi),ma))),deg,list_update(vEBT_VEBT,treeList,vEBT_VEBT_high(mi,na),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(mi,na)),vEBT_VEBT_low(mi,na))),
      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(mi,na))),vEBT_vebt_insert(summary,vEBT_VEBT_high(mi,na)),summary)) ).

% \<open>vebt_insert (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (x, max mi ma)) deg (treeList[high mi n := vebt_insert (treeList ! high mi n) (low mi n)]) (if minNull (treeList ! high mi n) then vebt_insert summary (high mi n) else summary)\<close>
tff(fact_563_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S),Ta)
       => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_564_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xb)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_565_pair__less__iff1,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Yb)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Z)),fun_pair_less)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Yb),Z) ) ).

% pair_less_iff1
tff(fact_566_bit__split__inv,axiom,
    ! [Xb: nat,D3: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xb,D3),vEBT_VEBT_low(Xb,D3),D3) = Xb ).

% bit_split_inv
tff(fact_567_list__update__overwrite,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,Xb: A,Yb: A] : list_update(A,list_update(A,Xs,I2,Xb),I2,Yb) = list_update(A,Xs,I2,Yb) ).

% list_update_overwrite
tff(fact_568_not__None__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ( Xb != none(A) )
    <=> ? [Y4: A] : Xb = aa(A,option(A),some(A),Y4) ) ).

% not_None_eq
tff(fact_569_not__Some__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ! [Y4: A] : Xb != aa(A,option(A),some(A),Y4)
    <=> ( Xb = none(A) ) ) ).

% not_Some_eq
tff(fact_570_list__update__id,axiom,
    ! [A: $tType,Xs: list(A),I2: nat] : list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),I2)) = Xs ).

% list_update_id
tff(fact_571_nth__list__update__neq,axiom,
    ! [A: $tType,I2: nat,J: nat,Xs: list(A),Xb: A] :
      ( ( I2 != J )
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,Xb)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_572_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_573_list__update__swap,axiom,
    ! [A: $tType,I2: nat,I5: nat,Xs: list(A),Xb: A,X6: A] :
      ( ( I2 != I5 )
     => ( list_update(A,list_update(A,Xs,I2,Xb),I5,X6) = list_update(A,list_update(A,Xs,I5,X6),I2,Xb) ) ) ).

% list_update_swap
tff(fact_574_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,nat,size_size(A),Xb) != aa(A,nat,size_size(A),Yb) )
         => ( Xb != Yb ) ) ) ).

% size_neq_size_imp_neq
tff(fact_575_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X2: A] : none(A) != aa(A,option(A),some(A),X2) ).

% option.distinct(1)
tff(fact_576_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X2: A] :
      ( ( Option = aa(A,option(A),some(A),X2) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_577_option_Oexhaust,axiom,
    ! [A: $tType,Yb: option(A)] :
      ( ( Yb != none(A) )
     => ~ ! [X22: A] : Yb != aa(A,option(A),some(A),X22) ) ).

% option.exhaust
tff(fact_578_split__option__ex,axiom,
    ! [A: $tType,P2: fun(option(A),$o)] :
      ( ? [X_13: option(A)] : aa(option(A),$o,P2,X_13)
    <=> ( aa(option(A),$o,P2,none(A))
        | ? [X4: A] : aa(option(A),$o,P2,aa(A,option(A),some(A),X4)) ) ) ).

% split_option_ex
tff(fact_579_split__option__all,axiom,
    ! [A: $tType,P2: fun(option(A),$o)] :
      ( ! [X_13: option(A)] : aa(option(A),$o,P2,X_13)
    <=> ( aa(option(A),$o,P2,none(A))
        & ! [X4: A] : aa(option(A),$o,P2,aa(A,option(A),some(A),X4)) ) ) ).

% split_option_all
tff(fact_580_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,Xb: option(A),P2: fun(option(A),fun(option(B),$o)),Yb: option(B)] :
      ( ( ( Xb = none(A) )
       => aa(option(B),$o,aa(option(A),fun(option(B),$o),P2,Xb),Yb) )
     => ( ( ( Yb = none(B) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P2,Xb),Yb) )
       => ( ! [A4: A,B4: B] :
              ( ( Xb = aa(A,option(A),some(A),A4) )
             => ( ( Yb = aa(B,option(B),some(B),B4) )
               => aa(option(B),$o,aa(option(A),fun(option(B),$o),P2,Xb),Yb) ) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P2,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_581_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_582_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : size_option(A,Xb,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_583_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A),Xb: A,I2: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A5)
     => ( member(A,Xb,A5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,Xb))),A5) ) ) ).

% set_update_subsetI
tff(fact_584_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_585_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_586_option_Osize__neq,axiom,
    ! [A: $tType,Xb: option(A)] : aa(option(A),nat,size_size(option(A)),Xb) != zero_zero(nat) ).

% option.size_neq
tff(fact_587_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X3: A,Y: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),R2)
         => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),S) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S) ) ).

% subrelI
tff(fact_588_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_589_pair__lessI1,axiom,
    ! [A2: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_590_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list(vEBT_VEBT),Xb: nat] :
      ( vEBT_V5917875025757280293ildren(Nb,TreeLista,Xb)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,Nb))),vEBT_VEBT_low(Xb,Nb)) ) ).

% in_children_def
tff(fact_591_inthall,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o),Nb: nat] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P2,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% inthall
tff(fact_592_pair__leqI2,axiom,
    ! [A2: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),S),Ta)
       => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_593_pair__leqI1,axiom,
    ! [A2: nat,B2: nat,S: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_594__C4_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I: nat] :
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m))
       => ( ( ( vEBT_VEBT_high(ma,na) = I )
           => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),vEBT_VEBT_low(ma,na)) )
          & ! [X: nat] :
              ( ( ( vEBT_VEBT_high(X,na) = I )
                & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),vEBT_VEBT_low(X,na)) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),mi),X)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),ma) ) ) ) ) ) ).

% "4.hyps"(9)
tff(fact_595_vebt__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X3)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),X3)
       => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),X3)
         => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X3)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_596_vebt__insert_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X3)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S3)),X3)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3)),X3)
         => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary)),X3)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_597_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Uw2)
     => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va3,Vb2)),X3)
         => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2)),X3)
           => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd)),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_598_intind,axiom,
    ! [A: $tType,I2: nat,Nb: nat,P2: fun(A,$o),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
     => ( aa(A,$o,P2,Xb)
       => aa(A,$o,P2,aa(nat,A,nth(A,replicate(A,Nb,Xb)),I2)) ) ) ).

% intind
tff(fact_599_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_600_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_601__C4_Ohyps_C_I2_J,axiom,
    aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m) ).

% "4.hyps"(2)
tff(fact_602__C4_Ohyps_C_I8_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),deg)) ).

% "4.hyps"(8)
tff(fact_603__C4_Oprems_C_I1_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),deg)) ).

% "4.prems"(1)
tff(fact_604__C4_Oprems_C_I2_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),ya),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),deg)) ).

% "4.prems"(2)
tff(fact_605_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_606__C4_OIH_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => ( vEBT_invar_vebt(X,na)
        & ! [Xa: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
           => ! [Xb2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
               => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(X,Xa)),Xb2)
                 => ( aa(nat,$o,vEBT_vebt_member(X),Xb2)
                    | ( Xa = Xb2 ) ) ) ) ) ) ) ).

% "4.IH"(1)
tff(fact_607__C4_Ohyps_C_I5_J,axiom,
    ! [I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m))
     => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),X_13)
      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(summary),I) ) ) ).

% "4.hyps"(5)
tff(fact_608_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Maa,Nb)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb)) ) ).

% high_bound_aux
tff(fact_609_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
     => ( vEBT_invar_vebt(Tree,Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ) ).

% member_bound
tff(fact_610_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ) ).

% numeral_le_iff
tff(fact_611_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ) ).

% numeral_less_iff
tff(fact_612_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ) ).

% numeral_plus_numeral
tff(fact_613_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V: num,W2: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W2)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W2))),Z) ) ).

% add_numeral_left
tff(fact_614_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_615_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_616_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Yb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Yb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Yb)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_617_replicate__eq__replicate,axiom,
    ! [A: $tType,Mb: nat,Xb: A,Nb: nat,Yb: A] :
      ( ( replicate(A,Mb,Xb) = replicate(A,Nb,Yb) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero(nat) )
         => ( Xb = Yb ) ) ) ) ).

% replicate_eq_replicate
tff(fact_618_length__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),replicate(A,Nb,Xb)) = Nb ).

% length_replicate
tff(fact_619_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I2,Xb)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_620_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_621__C4_OIH_C_I2_J,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Yb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m))
       => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(summary,Xb)),Yb)
         => ( aa(nat,$o,vEBT_vebt_member(summary),Yb)
            | ( Xb = Yb ) ) ) ) ) ).

% "4.IH"(2)
tff(fact_622_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_623_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xb)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(4)
tff(fact_624_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(3)
tff(fact_625_xyprop,axiom,
    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(xa,na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(ya,na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na)) ) ).

% xyprop
tff(fact_626__092_060open_062low_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ay_An_A_060_A2_A_094_An_092_060close_062,axiom,
    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(xa,na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(ya,na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na)) ) ).

% \<open>low x n < 2 ^ n \<and> low y n < 2 ^ n\<close>
tff(fact_627_Ball__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P2: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
         => aa(A,$o,P2,X4) )
    <=> ( aa(A,$o,P2,A2)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_628_Bex__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P2: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
          & aa(A,$o,P2,X4) )
    <=> ( aa(A,$o,P2,A2)
        & ( Nb != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_629_in__set__replicate,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Yb: A] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),replicate(A,Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_630_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2)
     => ( list_update(A,Xs,I2,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_631_nth__replicate,axiom,
    ! [A: $tType,I2: nat,Nb: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
     => ( aa(nat,A,nth(A,replicate(A,Nb,Xb)),I2) = Xb ) ) ).

% nth_replicate
tff(fact_632__C00_C,axiom,
    ( ( deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) )
    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),na)
    & ( na = m )
    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),deg)
    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na) ) ) ).

% "00"
tff(fact_633_nth__list__update__eq,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,Xb)),I2) = Xb ) ) ).

% nth_list_update_eq
tff(fact_634_mimaxyprop,axiom,
    ( ~ ( ( xa = mi )
        | ( xa = ma ) )
    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(xa,na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(mi,na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(xa,na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(mi,na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na) ) ) ).

% mimaxyprop
tff(fact_635_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) = aa(nat,nat,suc,aa(nat,nat,suc,Nb)) ).

% add_2_eq_Suc
tff(fact_636_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_637_set__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I2))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_638_Ex__list__of__length,axiom,
    ! [A: $tType,Nb: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = Nb ).

% Ex_list_of_length
tff(fact_639_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_640_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y: A] :
            ( member(A,Y,aa(list(A),set(A),set2(A),Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate(A,Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_641_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( X3 = Xb ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_642_less__by__empty,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
      ( ( A5 = bot_bot(set(product_prod(A,A))) )
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),A5),B5) ) ).

% less_by_empty
tff(fact_643_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),bit0(Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% numeral_Bit0
tff(fact_644_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),bit0(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_645_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X222))) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_646_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_647_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,Nb)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_648_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(Xb,Nb)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_649_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( ( Nb = zero_zero(nat) )
        | ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_650_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))
    <=> ( ( Nb = zero_zero(nat) )
        | ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_651_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_652_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),one2))),Nb) ).

% Suc_nat_number_of_add
tff(fact_653_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,D2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),D2)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList,Summary)),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_654_VEBT_Oexhaust,axiom,
    ! [Yb: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Yb != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Yb != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_655_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_656_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),Nb) ) ).

% zero_neq_numeral
tff(fact_657_finite__maxlen,axiom,
    ! [A: $tType,M4: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M4)
     => ? [N2: nat] :
        ! [X: list(A)] :
          ( member(list(A),X,M4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),N2) ) ) ).

% finite_maxlen
tff(fact_658_length__induct,axiom,
    ! [A: $tType,P2: fun(list(A),$o),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(list(A),$o,P2,Ys2) )
         => aa(list(A),$o,P2,Xs2) )
     => aa(list(A),$o,P2,Xs) ) ).

% length_induct
tff(fact_659_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X3,Nb) )
     => ( vEBT_invar_vebt(Summarya,Mb)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_660_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv)),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_661_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_662_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_663_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_664_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X3,Nb) )
     => ( vEBT_invar_vebt(Summarya,Mb)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb) )
         => ( ( Mb = aa(nat,nat,suc,Nb) )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_665_num_Osize_I5_J,axiom,
    ! [X2: num] : aa(num,nat,size_size(num),bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_666_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_le_numeral
tff(fact_667_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_668_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_less_numeral
tff(fact_669_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_670_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_671_Skolem__list__nth,axiom,
    ! [A: $tType,K2: nat,P2: fun(nat,fun(A,$o))] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K2)
         => ? [X_13: A] : aa(A,$o,aa(nat,fun(A,$o),P2,I4),X_13) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K2 )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K2)
             => aa(A,$o,aa(nat,fun(A,$o),P2,I4),aa(nat,A,nth(A,Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_672_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_673_set__encode__eq,axiom,
    ! [A5: set(nat),B5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A5)
     => ( aa(set(nat),$o,finite_finite2(nat),B5)
       => ( ( aa(set(nat),nat,nat_set_encode,A5) = aa(set(nat),nat,nat_set_encode,B5) )
        <=> ( A5 = B5 ) ) ) ) ).

% set_encode_eq
tff(fact_674_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_675_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X3,Nb) )
     => ( vEBT_invar_vebt(Summarya,Mb)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,Nb) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(X3,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),X3)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Maa) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_676_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: $o,B4: $o] : A1 = vEBT_Leaf((A4),(B4))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList: list(vEBT_VEBT),N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => vEBT_invar_vebt(X,N2) )
                 => ( vEBT_invar_vebt(Summary,M)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                     => ( ( M = N2 )
                       => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M) )
                         => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list(vEBT_VEBT),N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => vEBT_invar_vebt(X,N2) )
                   => ( vEBT_invar_vebt(Summary,M)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                       => ( ( M = aa(nat,nat,suc,N2) )
                         => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M) )
                           => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list(vEBT_VEBT),N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_invar_vebt(X,N2) )
                     => ( vEBT_invar_vebt(Summary,M)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                         => ( ( M = N2 )
                           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M) )
                             => ( ! [I: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),X_13)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I: nat] :
                                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                                             => ( ( ( vEBT_VEBT_high(Ma,N2) = I )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),vEBT_VEBT_low(Ma,N2)) )
                                                & ! [X: nat] :
                                                    ( ( ( vEBT_VEBT_high(X,N2) = I )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),vEBT_VEBT_low(X,N2)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list(vEBT_VEBT),N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_invar_vebt(X,N2) )
                       => ( vEBT_invar_vebt(Summary,M)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
                           => ( ( M = aa(nat,nat,suc,N2) )
                             => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M) )
                               => ( ! [I: nat] :
                                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                                     => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),X_13)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                         => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                                               => ( ( ( vEBT_VEBT_high(Ma,N2) = I )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),vEBT_VEBT_low(Ma,N2)) )
                                                  & ! [X: nat] :
                                                      ( ( ( vEBT_VEBT_high(X,N2) = I )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),vEBT_VEBT_low(X,N2)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_677_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A7: $o,B7: $o] : A1 = vEBT_Leaf((A7),(B7))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X4,N) )
            & vEBT_invar_vebt(Summary2,N)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X4,N) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X4,N) )
            & vEBT_invar_vebt(Summary2,N)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
                 => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N)) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X4,N)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X4)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X4,N) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N)) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X4,N)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X4)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_678_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),Xb)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% count_le_length
tff(fact_679_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X3,Nb) )
     => ( vEBT_invar_vebt(Summarya,Mb)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb) )
         => ( ( Mb = aa(nat,nat,suc,Nb) )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,Nb) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(X3,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),X3)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Maa) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_680_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X3)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Ux2)
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S3: vEBT_VEBT,X3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_681_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_682_length__pos__if__in__set,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_683_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_684_nth__mem,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,aa(nat,A,nth(A,Xs),Nb),aa(list(A),set(A),set2(A),Xs)) ) ).

% nth_mem
tff(fact_685_list__ball__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P2: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P2,X3) )
       => aa(A,$o,P2,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_686_in__set__conv__nth,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_687_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o),Xb: A] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P2,aa(nat,A,nth(A,Xs),I3)) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P2,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_688_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X4) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P2,aa(nat,A,nth(A,Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_689_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_690_set__update__memI,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,Xb,aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_691_nth__list__update,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Xb: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,Xb)),J) = $ite(I2 = J,Xb,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_692_list__update__same__conv,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I2,Xb) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I2) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_693_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_694_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_695_set__encode__inf,axiom,
    ! [A5: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),A5)
     => ( aa(set(nat),nat,nat_set_encode,A5) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_696_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Yb: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Yb) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Yb) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Yb) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Yb) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ (Yb) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => (Yb) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_697_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Yb = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_698_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_699_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
           => ( ( aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( Xb = Yb ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_700_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_701_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Mia,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mia),Maa))),Dega,list_update(vEBT_VEBT,TreeLista,vEBT_VEBT_high(Mia,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Mia,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mia,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Mia,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Mia,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_702_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xb),Maa))),Dega,list_update(vEBT_VEBT,TreeLista,vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_703_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_704_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,B2),Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_705_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(nat,A,power_power(A,A2),Nb) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% power_eq_0_iff
tff(fact_706_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Maa)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_707_pow__sum,axiom,
    ! [A2: nat,B2: nat] : divide_divide(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B2) ).

% pow_sum
tff(fact_708_high__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_high(Xb,Nb) = divide_divide(nat,Xb,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% high_def
tff(fact_709_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_710_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( divide_divide(A,A2,C3) = divide_divide(A,B2,C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_711_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( divide_divide(A,C3,A2) = divide_divide(A,C3,B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_712_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_713_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_714_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% div_0
tff(fact_715_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : aa(nat,A,power_power(A,zero_zero(A)),aa(nat,nat,suc,Nb)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_716_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K2: num] : aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),K2)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_717_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_718_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa(nat,nat,power_power(nat,Xb),Mb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Mb = zero_zero(nat) )
        | ( Xb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_719_power__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_720_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,power_power(nat,Xb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_721_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_722_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ).

% add_divide_distrib
tff(fact_723_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),divide_divide(A,A2,C3)) ) ) ) ).

% divide_right_mono_neg
tff(fact_724_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xb,Yb)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_725_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Yb)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_726_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Yb)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_727_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xb,Yb)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_728_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_729_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% divide_right_mono
tff(fact_730_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_731_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_732_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% divide_strict_right_mono
tff(fact_733_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A2,B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_734_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( C3 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_735_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_less_0_iff
tff(fact_736_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,Xb,Yb)) ) ) ) ).

% divide_pos_pos
tff(fact_737_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Yb)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_738_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Yb)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_739_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,Xb,Yb)) ) ) ) ).

% divide_neg_neg
tff(fact_740_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Yb)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_741_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xb,Yb)) ) ) ) ).

% divide_nonpos_neg
tff(fact_742_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xb,Yb)) ) ) ) ).

% divide_nonneg_pos
tff(fact_743_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Yb)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_744_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_745_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Z)),divide_divide(A,Yb,W2)) ) ) ) ) ) ).

% frac_less2
tff(fact_746_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Z)),divide_divide(A,Yb,W2)) ) ) ) ) ) ).

% frac_less
tff(fact_747_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Yb: A,Xb: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Z)),divide_divide(A,Yb,W2)) ) ) ) ) ) ).

% frac_le
tff(fact_748_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,aa(num,A,numeral_numeral(A),bit0(one2)))),divide_divide(A,Xb,aa(num,A,numeral_numeral(A),bit0(one2)))) = Xb ) ).

% field_sum_of_halves
tff(fact_749_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_750_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_751_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_752_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Nb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,power_power(A,A2),Nb) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_753_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Nb)) ) ) ).

% zero_le_power
tff(fact_754_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,B2),Nb)) ) ) ) ).

% power_mono
tff(fact_755_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Nb)) ) ) ).

% zero_less_power
tff(fact_756_nat__power__less__imp__less,axiom,
    ! [I2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,I2),Mb)),aa(nat,nat,power_power(nat,I2),Nb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_757_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,B2),Nb))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_758_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Nb))),aa(nat,A,power_power(A,B2),aa(nat,nat,suc,Nb)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_759_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Nb)) = aa(nat,A,power_power(A,B2),aa(nat,nat,suc,Nb)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_760_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(nat,A,power_power(A,zero_zero(A)),Nb) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_761_power__gt__expt,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(nat,nat,power_power(nat,Nb),K2)) ) ).

% power_gt_expt
tff(fact_762_nat__one__le__power,axiom,
    ! [I2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,power_power(nat,I2),Nb)) ) ).

% nat_one_le_power
tff(fact_763_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,power_power(A,A2),Nb) = aa(nat,A,power_power(A,B2),Nb) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_764_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( ( aa(nat,A,power_power(A,A2),Nb) = aa(nat,A,power_power(A,B2),Nb) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_765_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_766_less__exp,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% less_exp
tff(fact_767_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,Mb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_768_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,Mb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,power_power(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_769_self__le__ge2__pow,axiom,
    ! [K2: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,power_power(nat,K2),Mb)) ) ).

% self_le_ge2_pow
tff(fact_770_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_771_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
             => ( Xb = Yb ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_772_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ) ).

% power2_le_imp_le
tff(fact_773_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,B2),Nb)) ) ) ) ) ).

% power_strict_mono
tff(fact_774_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_775_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb) ) ) ) ).

% power2_less_imp_less
tff(fact_776_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Yb = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_777_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_power2_ge_zero
tff(fact_778_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( ( Xb != zero_zero(A) )
            | ( Yb != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_779_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_780_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Dega)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_781_add__self__div__2,axiom,
    ! [Mb: nat] : divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Mb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Mb ).

% add_self_div_2
tff(fact_782_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Dega)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_783_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit0(Nb)) ).

% semiring_norm(76)
tff(fact_784_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( divide_divide(nat,Mb,Nb) = zero_zero(nat) ) ) ).

% div_less
tff(fact_785_div__by__Suc__0,axiom,
    ! [Mb: nat] : divide_divide(nat,Mb,aa(nat,nat,suc,zero_zero(nat))) = Mb ).

% div_by_Suc_0
tff(fact_786_set__n__deg__not__0,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Mb: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X3,Nb) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_787_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_788_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_789_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Mb: nat,Nb: nat] : divide_divide(A,divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ).

% div_exp_eq
tff(fact_790_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A7: $o,B7: $o] : Ta = vEBT_Leaf((A7),(B7)) ) ).

% deg1Leaf
tff(fact_791_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
     => ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ).

% deg_1_Leaf
tff(fact_792_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( Nb = one_one(nat) )
       => ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ) ).

% deg_1_Leafy
tff(fact_793_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_794_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_795_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_796_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,A2,one_one(A)) = A2 ) ).

% div_by_1
tff(fact_797_semiring__norm_I78_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(Mb)),bit0(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ).

% semiring_norm(78)
tff(fact_798_semiring__norm_I75_J,axiom,
    ! [Mb: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),one2) ).

% semiring_norm(75)
tff(fact_799_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_800_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_801_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_802_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_803_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          divide_divide(A,A2,A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_804_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_805_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_806_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_807_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_808_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( ( aa(nat,A,power_power(A,A2),Mb) = aa(nat,A,power_power(A,A2),Nb) )
          <=> ( Mb = Nb ) ) ) ) ).

% power_inject_exp
tff(fact_809_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_810_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_811_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(6)
tff(fact_812_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(5)
tff(fact_813_less__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),one_one(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_one
tff(fact_814_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_815_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_816_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_817_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_818_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_819_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_820_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_821_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_822_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Yb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,B2),Xb)),aa(nat,A,power_power(A,B2),Yb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Yb) ) ) ) ).

% power_strict_increasing_iff
tff(fact_823_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_824_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_825_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_826_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_827_one__add__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% one_add_one
tff(fact_828_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,B2),Mb)),aa(nat,A,power_power(A,B2),Nb))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_829_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Yb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,B2),Xb)),aa(nat,A,power_power(A,B2),Yb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Yb) ) ) ) ).

% power_increasing_iff
tff(fact_830_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb)) ) ).

% one_plus_numeral
tff(fact_831_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ) ).

% numeral_plus_one
tff(fact_832_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),one2) ) ) ).

% numeral_le_one_iff
tff(fact_833_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),Nb) ) ) ).

% one_less_numeral_iff
tff(fact_834_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_835_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_836_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,B2),Mb)),aa(nat,A,power_power(A,B2),Nb))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ) ) ) ) ).

% power_decreasing_iff
tff(fact_837_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xb: A] :
          ( ( one_one(A) = Xb )
        <=> ( Xb = one_one(A) ) ) ) ).

% one_reorient
tff(fact_838_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_839_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_840_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_841_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_842_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_843_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_844_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_845_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_846_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_847_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_848_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_849_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_le_numeral
tff(fact_850_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_851_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))) ) ).

% less_add_one
tff(fact_852_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_853_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_854_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_855_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,power_power(A,A2),Nb)) ) ) ).

% one_le_power
tff(fact_856_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Mb,Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_857_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( ( divide_divide(nat,Mb,Nb) = Mb )
      <=> ( Nb = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_858_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_859_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_860_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),Nb) ).

% Suc_eq_plus1_left
tff(fact_861_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_862_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_863_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D3: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv)),D3)
    <=> ( D3 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_864_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))) ) ).

% zero_less_two
tff(fact_865_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Nb)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_866_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_867_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% less_divide_eq_1
tff(fact_868_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),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_869_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_870_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(nat,A,power_power(A,zero_zero(A)),Nb) = $ite(Nb = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_871_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Nb))) ) ) ).

% power_gt1
tff(fact_872_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Mb)),aa(nat,A,power_power(A,A2),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% power_less_imp_less_exp
tff(fact_873_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,A2),N3)) ) ) ) ).

% power_strict_increasing
tff(fact_874_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,A2),N3)) ) ) ) ).

% power_increasing
tff(fact_875_nat__induct__non__zero,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,P2,one_one(nat))
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => ( aa(nat,$o,P2,N2)
               => aa(nat,$o,P2,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,P2,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_876_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A2),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A2),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_877_vebt__insert_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A2),(B2)),Xb) = $ite(
        Xb = zero_zero(nat),
        vEBT_Leaf($true,(B2)),
        $ite(Xb = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_878_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A2),(B2)),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A2),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_879_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_880_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_881_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Nb))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_882_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,Nb))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_883_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),N3)),aa(nat,A,power_power(A,A2),Nb)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_884_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),N3)),aa(nat,A,power_power(A,A2),Nb)) ) ) ) ) ).

% power_decreasing
tff(fact_885_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Mb)),aa(nat,A,power_power(A,A2),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% power_le_imp_le_exp
tff(fact_886_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,power_power(A,A2),Nb)) ) ) ) ).

% self_le_power
tff(fact_887_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A2),Nb)) ) ) ) ).

% one_less_power
tff(fact_888_nat__1__add__1,axiom,
    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% nat_1_add_1
tff(fact_889_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Mb,K2)),divide_divide(nat,Nb,K2)) ) ).

% div_le_mono
tff(fact_890_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Mb,Nb)),Mb) ).

% div_le_dividend
tff(fact_891_ex__power__ivl2,axiom,
    ! [B2: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)
       => ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),N2)),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(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_892_ex__power__ivl1,axiom,
    ! [B2: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K2)
       => ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),N2)),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(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_893_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( divide_divide(nat,Mb,Nb) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_894_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Mb,Nb)),divide_divide(nat,aa(nat,nat,suc,Mb),Nb)) ).

% Suc_div_le_mono
tff(fact_895_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,K2,Nb)),divide_divide(nat,K2,Mb)) ) ) ).

% div_le_mono2
tff(fact_896_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,Mb,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_897_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_898_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_899_nat__induct2,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( aa(nat,$o,P2,one_one(nat))
       => ( ! [N2: nat] :
              ( aa(nat,$o,P2,N2)
             => aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P2,Nb) ) ) ) ).

% nat_induct2
tff(fact_900_bit__concat__def,axiom,
    ! [H: nat,L: nat,D3: nat] : vEBT_VEBT_bit_concat(H,L,D3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),D3))),L) ).

% bit_concat_def
tff(fact_901_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Yb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),Xb),Nb) = Xb ) ) ).

% low_inv
tff(fact_902_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Yb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),Xb),Nb) = Yb ) ) ).

% high_inv
tff(fact_903_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),Mb)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(1)
tff(fact_904_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),Mb)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(2)
tff(fact_905_pos2,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_906_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2) ) ) ).

% discrete
tff(fact_907_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_908_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A2,B2)) ) ) ) ).

% div_positive
tff(fact_909_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ).

% zle_add1_eq_le
tff(fact_910_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Nb)
    <=> ( Nb != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_911_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_912_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_913_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_914_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_915_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_916_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_917_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_918_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C3: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C3)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C3) ) ).

% times_divide_eq_left
tff(fact_919_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,divide_divide(A,A2,B2),C3) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% divide_divide_eq_left
tff(fact_920_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,A2,divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),B2) ) ).

% divide_divide_eq_right
tff(fact_921_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ).

% times_divide_eq_right
tff(fact_922_mult__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_923_mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_924_mult__0__right,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_925_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) = zero_zero(nat) )
    <=> ( ( Mb = zero_zero(nat) )
        | ( Nb = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_926_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) )
    <=> ( ( Mb = one_one(nat) )
        & ( Nb = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_927_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) = one_one(nat) )
    <=> ( ( Mb = one_one(nat) )
        & ( Nb = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_928_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_929_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B2: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_930_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_931_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B2: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_932_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Yb)) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Yb = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_933_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_934_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_935_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_936_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_937_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_938_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_939_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = $ite(C3 = zero_zero(A),zero_zero(A),divide_divide(A,A2,B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_940_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = $ite(C3 = zero_zero(A),zero_zero(A),divide_divide(A,A2,B2)) ) ).

% div_mult_mult1_if
tff(fact_941_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult2
tff(fact_942_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult1
tff(fact_943_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_944_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C3)) ) ).

% distrib_left_numeral
tff(fact_945_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Mb = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_946_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) )
    <=> ( ( Mb = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_947_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_948_mult__less__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_949_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_950_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ).

% mult_Suc_right
tff(fact_951_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) = $ite(K2 = zero_zero(nat),zero_zero(nat),divide_divide(nat,Mb,Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_952_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K2)
       => ( divide_divide(int,K2,L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_953_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),L)
       => ( divide_divide(int,K2,L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_954_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_955_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_956_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 )
        <=> $ite(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)),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_957_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)) )
        <=> $ite(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,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_958_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_959_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))),B2) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_960_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_961_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_962_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: 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),C3)),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self4
tff(fact_963_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: 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),C3),B2)),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self3
tff(fact_964_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C3: 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),C3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self2
tff(fact_965_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C3: 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),C3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self1
tff(fact_966_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) ) ) ).

% one_le_mult_iff
tff(fact_967_mult__le__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_968_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_969_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Mb),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_970_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_971_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ).

% add1_zle_eq
tff(fact_972_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ).

% le_imp_0_less
tff(fact_973_zdiv__mono1,axiom,
    ! [A2: int,A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B2)),divide_divide(int,A3,B2)) ) ) ).

% zdiv_mono1
tff(fact_974_zdiv__mono2,axiom,
    ! [A2: int,B3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B2)),divide_divide(int,A2,B3)) ) ) ) ).

% zdiv_mono2
tff(fact_975_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( divide_divide(int,I2,K2) = zero_zero(int) )
    <=> ( ( K2 = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K2) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),I2) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_976_zdiv__mono1__neg,axiom,
    ! [A2: int,A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A3,B2)),divide_divide(int,A2,B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_977_zdiv__mono2__neg,axiom,
    ! [A2: int,B3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B3)),divide_divide(int,A2,B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_978_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_979_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,K2,L))
    <=> ( ( K2 = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_980_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),divide_divide(int,K2,L)) ) ) ).

% div_positive_int
tff(fact_981_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_982_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_983_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),divide_divide(int,I2,K2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),I2) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_984_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_985_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,A2,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_986_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,A2,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_987_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),divide_divide(int,A2,B2))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_988_odd__nonzero,axiom,
    ! [Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z) != zero_zero(int) ).

% odd_nonzero
tff(fact_989_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_990_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_991_int__div__less__self,axiom,
    ! [Xb: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,Xb,K2)),Xb) ) ) ).

% int_div_less_self
tff(fact_992_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_993_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_994_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_995_int__gr__induct,axiom,
    ! [K2: int,I2: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),I2)
     => ( aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),I3)
             => ( aa(int,$o,P2,I3)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
         => aa(int,$o,P2,I2) ) ) ) ).

% int_gr_induct
tff(fact_996_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_997_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% mult.assoc
tff(fact_998_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_999_mult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% mult.left_commute
tff(fact_1000_less__int__code_I1_J,axiom,
    ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_1001_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_1002_enat__less__induct,axiom,
    ! [P2: fun(extended_enat,$o),Nb: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M3: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M3),N2)
             => aa(extended_enat,$o,P2,M3) )
         => aa(extended_enat,$o,P2,N2) )
     => aa(extended_enat,$o,P2,Nb) ) ).

% enat_less_induct
tff(fact_1003_plus__int__code_I1_J,axiom,
    ! [K2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),zero_zero(int)) = K2 ).

% plus_int_code(1)
tff(fact_1004_plus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),L) = L ).

% plus_int_code(2)
tff(fact_1005_less__eq__int__code_I1_J,axiom,
    aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_1006_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_1007_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_1008_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_1009_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_1010_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_1011_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_1012_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_1013_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( ( A2 != B2 )
            & ( C3 != D3 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% crossproduct_noteq
tff(fact_1014_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W2: A,Yb: A,Xb: A,Z: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Yb)),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb)) )
        <=> ( ( W2 = Xb )
            | ( Yb = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_1015_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E3: A,B2: A,C3: 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)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),E3)),C3) ) ).

% combine_common_factor
tff(fact_1016_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% distrib_right
tff(fact_1017_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ).

% distrib_left
tff(fact_1018_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% comm_semiring_class.distrib
tff(fact_1019_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ).

% ring_class.ring_distribs(1)
tff(fact_1020_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% ring_class.ring_distribs(2)
tff(fact_1021_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Yb: A,Z: A,W2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,Xb,Yb)),divide_divide(A,Z,W2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),W2)) ) ).

% times_divide_times_eq
tff(fact_1022_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Yb: A,Z: A,W2: A] : divide_divide(A,divide_divide(A,Xb,Yb),divide_divide(A,Z,W2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),W2),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z)) ) ).

% divide_divide_times_eq
tff(fact_1023_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,divide_divide(A,A2,B2),C3) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).

% divide_divide_eq_left'
tff(fact_1024_Suc__mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_1025_mult__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),Nb) = zero_zero(nat) ).

% mult_0
tff(fact_1026_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb) )
    <=> ( ( K2 = zero_zero(nat) )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1027_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),J)) ) ).

% mult_le_mono2
tff(fact_1028_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K2)) ) ).

% mult_le_mono1
tff(fact_1029_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_1030_le__square,axiom,
    ! [Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Mb)) ).

% le_square
tff(fact_1031_le__cube,axiom,
    ! [Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Mb))) ).

% le_cube
tff(fact_1032_add__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) ).

% add_mult_distrib2
tff(fact_1033_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2)) ).

% add_mult_distrib
tff(fact_1034_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),U)),K2) ).

% left_add_mult_distrib
tff(fact_1035_nat__mult__1,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),Nb) = Nb ).

% nat_mult_1
tff(fact_1036_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),one_one(nat)) = Nb ).

% nat_mult_1_right
tff(fact_1037_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)) ).

% nat_mult_max_right
tff(fact_1038_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ).

% nat_mult_max_left
tff(fact_1039_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A2,B2),C3) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1040_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1041_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1042_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1043_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_1044_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_1045_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_1046_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_1047_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1048_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_right_mono
tff(fact_1049_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_right_mono_neg
tff(fact_1050_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_left_mono
tff(fact_1051_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_1052_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1053_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ).

% split_mult_pos_le
tff(fact_1054_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)) ) ).

% zero_le_square
tff(fact_1055_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1056_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_mono
tff(fact_1057_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1058_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1059_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_strict_right_mono
tff(fact_1060_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1061_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1062_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_1063_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1064_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1065_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1066_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1067_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_1068_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1069_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_1070_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_pos_pos
tff(fact_1071_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_1072_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_1073_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1074_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),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_1075_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_neg_neg
tff(fact_1076_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C3: A,D3: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C3 != D3 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1077_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Mb: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),Nb)) ) ) ) ).

% less_1_mult
tff(fact_1078_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Yb: A,Z: A,Xb: A,W2: A] :
          ( ( Yb != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( divide_divide(A,Xb,Yb) = divide_divide(A,W2,Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Yb) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1079_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( divide_divide(A,B2,C3) = A2 )
        <=> $ite(C3 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_1080_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = divide_divide(A,B2,C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_1081_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) )
           => ( divide_divide(A,B2,C3) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_1082_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2 )
           => ( A2 = divide_divide(A,B2,C3) ) ) ) ) ).

% eq_divide_imp
tff(fact_1083_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( divide_divide(A,B2,C3) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1084_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( A2 = divide_divide(A,B2,C3) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1085_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_1086_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Mb: nat,Nb: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),Mb)),aa(nat,A,power_power(A,A2),Nb)) ) ).

% power_add
tff(fact_1087_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),J)) ) ) ).

% mult_less_mono2
tff(fact_1088_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K2)) ) ) ).

% mult_less_mono1
tff(fact_1089_nat__mult__less__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_1090_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1091_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_1092_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ).

% mult_Suc
tff(fact_1093_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) )
     => ( ( Nb = one_one(nat) )
        | ( Mb = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1094_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Mb,Nb)),I2) ) ).

% less_mult_imp_div_less
tff(fact_1095_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,Mb,Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_1096_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),divide_divide(nat,Mb,Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_1097_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1098_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1099_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1100_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1101_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1102_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1103_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1104_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1105_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1106_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1107_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1108_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1109_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1110_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1111_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),A2) ) ) ) ).

% mult_left_le
tff(fact_1112_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_1113_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb)),Xb) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1114_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Xb)),Xb) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1115_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Yb))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Yb = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1116_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Yb))) ) ).

% sum_squares_ge_zero
tff(fact_1117_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Yb)))
        <=> ( ( Xb != zero_zero(A) )
            | ( Yb != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1118_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Yb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Yb))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1119_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C3,A2)),divide_divide(A,C3,B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1120_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C3,A2)),divide_divide(A,C3,B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1121_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Yb: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Yb)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),divide_divide(A,Xb,Yb)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1122_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Yb: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Yb))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Yb)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1123_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_1124_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1125_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1126_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_1127_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1128_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_1129_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,W2: num] :
          ( ( divide_divide(A,B2,C3) = aa(num,A,numeral_numeral(A),W2) )
        <=> $ite(C3 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3),aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1130_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C3: A] :
          ( ( aa(num,A,numeral_numeral(A),W2) = divide_divide(A,B2,C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3) = B2,aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1131_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Z)),Yb) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z)),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1132_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,Yb,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),Xb),Z)),Yb),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1133_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Yb: A,Z: A,Xb: A] :
          ( ( Yb != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,Xb,Yb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Yb)),Yb) ) ) ) ).

% add_num_frac
tff(fact_1134_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Yb: A,Xb: A,Z: A] :
          ( ( Yb != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Yb)),Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Yb)),Yb) ) ) ) ).

% add_frac_num
tff(fact_1135_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Yb: A,Z: A,Xb: A,W2: A] :
          ( ( Yb != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Yb)),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),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Yb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1136_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),divide_divide(A,B2,Z)) = $ite(Z = zero_zero(A),A2,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_1137_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Z)),B2) = $ite(Z = zero_zero(A),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_1138_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),Nb))) ) ) ).

% power_less_power_Suc
tff(fact_1139_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),Nb))) ) ) ).

% power_gt1_lemma
tff(fact_1140_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_1141_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_1142_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_1143_nat__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_1144_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Mb,Q2)),Nb)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_1145_nat__mult__div__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) = divide_divide(nat,Mb,Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1146_realpow__pos__nth2,axiom,
    ! [A2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ? [R3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
          & ( aa(nat,real,power_power(real,R3),aa(nat,nat,suc,Nb)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1147_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] :
          ( ! [Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Xb)),Yb) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ).

% field_le_mult_one_interval
tff(fact_1148_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1149_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1150_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1151_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1152_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1153_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1154_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1155_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1156_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C3,A2)),divide_divide(A,C3,B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1157_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Yb: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Yb)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),divide_divide(A,Xb,Yb)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1158_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Yb: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Yb))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Yb)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1159_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_1160_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1161_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1162_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_1163_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C3,A2)),divide_divide(A,C3,B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1164_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1165_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_1166_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xb: A,A2: A,Yb: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Yb))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1167_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1168_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1169_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),Nb))),aa(nat,A,power_power(A,A2),Nb)) ) ) ) ).

% power_Suc_less
tff(fact_1170_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2
tff(fact_1171_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2_right
tff(fact_1172_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),B2) ) ).

% left_add_twice
tff(fact_1173_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q2)))
       => ( divide_divide(nat,Mb,Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_1174_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),divide_divide(nat,Nb,Q2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1175_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),divide_divide(nat,Mb,Nb)))) ) ).

% dividend_less_times_div
tff(fact_1176_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,Mb,Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_1177_split__div,axiom,
    ! [P2: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P2,divide_divide(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P2,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I4)),J3) )
               => aa(nat,$o,P2,I4) ) ) ) ) ) ).

% split_div
tff(fact_1178_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xb: A,A2: A,Yb: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Yb))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1179_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1180_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1181_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)),Yb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_squares_bound
tff(fact_1182_split__div_H,axiom,
    ! [P2: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P2,divide_divide(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
          & aa(nat,$o,P2,zero_zero(nat)) )
        | ? [Q5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q5)),Mb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q5)))
            & aa(nat,$o,P2,Q5) ) ) ) ).

% split_div'
tff(fact_1183_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Yb: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)),Yb)) ) ).

% power2_sum
tff(fact_1184_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ) ).

% zero_le_even_power'
tff(fact_1185_nat__bit__induct,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,P2,N2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa(nat,$o,P2,N2)
             => aa(nat,$o,P2,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))) )
         => aa(nat,$o,P2,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_1186_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xb: A,Yb: A] :
          ( ( aa(nat,A,power_power(A,U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1187_triangle__def,axiom,
    ! [Nb: nat] : nat_triangle(Nb) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% triangle_def
tff(fact_1188_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X3)
            & ( aa(nat,real,power_power(real,X3),Nb) = A2 )
            & ! [Y3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y3)
                  & ( aa(nat,real,power_power(real,Y3),Nb) = A2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1189_realpow__pos__nth,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ( aa(nat,real,power_power(real,R3),Nb) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1190_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1191_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(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),bit0(one2))),Nb)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_1192_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),bit0(one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_1193_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),bit0(one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_1194_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Yb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1195_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1196_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_1197_low__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_low(Xb,Nb) = modulo_modulo(nat,Xb,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% low_def
tff(fact_1198_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1199_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( member(nat,aa(nat,nat,suc,Nb),nat_set_decode(Xb))
    <=> member(nat,Nb,nat_set_decode(divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_1200_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Yb )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Yb != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B4)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S3) )
             => ( Yb != vEBT_Node(Info2,zero_zero(nat),Ts2,S3) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3) )
               => ( Yb != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3) ) )
           => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) )
                 => ( Yb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                   => ( Yb != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),
                                    aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Xaa,Mi)),
                                    aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_1201_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_1202_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),one_one(nat))
    <=> ( Mb = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_1203_finite__Collect__disjI,axiom,
    ! [A: $tType,P2: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q)))
    <=> ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
        & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% finite_Collect_disjI
tff(fact_1204_finite__Collect__conjI,axiom,
    ! [A: $tType,P2: fun(A,$o),Q: fun(A,$o)] :
      ( ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
        | aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) )
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q))) ) ).

% finite_Collect_conjI
tff(fact_1205_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ac(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_1206_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ad(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_1207_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),zero_zero(A)) ) ).

% dvd_0_right
tff(fact_1208_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),A2)
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1209_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1210_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1211_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Mb = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_1212_dvd__1__left,axiom,
    ! [K2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K2) ).

% dvd_1_left
tff(fact_1213_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_1214_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_1215_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_1216_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_1217_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,B2,A2)),divide_divide(A,C3,A2))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3) ) ) ) ) ).

% div_dvd_div
tff(fact_1218_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Xb))
    <=> ( Xb = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1219_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_add_self1
tff(fact_1220_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_add_self2
tff(fact_1221_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
    <=> ( ( K2 = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1222_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2638667681897837118et_bit(int,Nb,K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_1223_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se5668285175392031749et_bit(int,Nb,K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_1224_mod__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( modulo_modulo(nat,Mb,Nb) = Mb ) ) ).

% mod_less
tff(fact_1225_finite__Collect__subsets,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ae(set(A),fun(set(A),$o),A5))) ) ).

% finite_Collect_subsets
tff(fact_1226_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_af(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_1227_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ag(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_1228_finite__Collect__less__nat,axiom,
    ! [K2: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),K2))) ).

% finite_Collect_less_nat
tff(fact_1229_finite__Collect__le__nat,axiom,
    ! [K2: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),K2))) ).

% finite_Collect_le_nat
tff(fact_1230_set__decode__inverse,axiom,
    ! [Nb: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(Nb)) = Nb ).

% set_decode_inverse
tff(fact_1231_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1232_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1233_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( C3 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1234_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( C3 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1235_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_1236_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1237_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1238_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_1239_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_1240_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_1241_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_1242_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_1243_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_1244_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( divide_divide(A,one_one(A),divide_divide(A,one_one(A),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_1245_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,one_one(A),A2)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_1246_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1247_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ) ).

% div_add
tff(fact_1248_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_1249_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_1250_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self1
tff(fact_1251_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self2
tff(fact_1252_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self3
tff(fact_1253_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self4
tff(fact_1254_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1255_mod__by__Suc__0,axiom,
    ! [Mb: nat] : modulo_modulo(nat,Mb,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_1256_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_1257_set__encode__inverse,axiom,
    ! [A5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A5)
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A5)) = A5 ) ) ).

% set_encode_inverse
tff(fact_1258_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,one_one(A),A2)) = divide_divide(A,B2,A2) ) ) ) ).

% unit_mult_div_div
tff(fact_1259_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_1260_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,B2),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1261_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self1
tff(fact_1262_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self2
tff(fact_1263_Suc__mod__mult__self3,axiom,
    ! [K2: nat,Nb: nat,Mb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self3
tff(fact_1264_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K2: nat,Mb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self4
tff(fact_1265_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% odd_add
tff(fact_1266_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_add
tff(fact_1267_set__decode__0,axiom,
    ! [Xb: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xb))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xb) ) ).

% set_decode_0
tff(fact_1268_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W2)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1269_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Nb)),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_1270_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1271_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_plus_one_iff
tff(fact_1272_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),bit0(one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1273_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),bit0(one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1274_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1275_add__self__mod__2,axiom,
    ! [Mb: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Mb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_1276_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W2)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1277_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1278_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1279_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1280_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,power_power(A,A2),Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% even_power
tff(fact_1281_mod2__gr__0,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_1282_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1283_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2))
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1284_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_1285_int__ge__induct,axiom,
    ! [K2: int,I2: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),I2)
     => ( aa(int,$o,P2,K2)
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),I3)
             => ( aa(int,$o,P2,I3)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
         => aa(int,$o,P2,I2) ) ) ) ).

% int_ge_induct
tff(fact_1286_real__arch__pow,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ? [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),aa(nat,real,power_power(real,Xb),N2)) ) ).

% real_arch_pow
tff(fact_1287_real__arch__pow__inv,axiom,
    ! [Yb: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ? [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,Xb),N2)),Yb) ) ) ).

% real_arch_pow_inv
tff(fact_1288_zdvd__mult__cancel,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),Mb)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),Nb))
     => ( ( K2 != zero_zero(int) )
       => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_1289_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% mod_0_imp_dvd
tff(fact_1290_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1291_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1292_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero(int) )
     => aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_aj(int,fun(int,$o),I2))) ) ).

% finite_divisors_int
tff(fact_1293_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ak(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ak(A,fun(A,$o),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1294_zdvd__reduce,axiom,
    ! [K2: int,Nb: int,Mb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K2),Mb)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K2),Nb) ) ).

% zdvd_reduce
tff(fact_1295_zdvd__period,axiom,
    ! [A2: int,D3: int,Xb: int,Ta: int,C3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),D3)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Ta))
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),C3),D3))),Ta)) ) ) ).

% zdvd_period
tff(fact_1296_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W2)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W2)) ).

% int_distrib(1)
tff(fact_1297_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z22)) ).

% int_distrib(2)
tff(fact_1298_less__set__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A5)),aTP_Lamp_a(set(A),fun(A,$o),B5)) ) ).

% less_set_def
tff(fact_1299_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_1300_less__eq__real__def,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Yb)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% less_eq_real_def
tff(fact_1301_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ak(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ak(A,fun(A,$o),B2)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_1302_empty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_al(A,$o)) ).

% empty_def
tff(fact_1303_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),R: fun(A,fun(B,$o))] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( ! [X3: A] :
              ( member(A,X3,A5)
             => ? [Xa: B] :
                  ( member(B,Xa,B5)
                  & aa(B,$o,aa(A,fun(B,$o),R,X3),Xa) ) )
         => ? [X3: B] :
              ( member(B,X3,B5)
              & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_am(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A5),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1304_not__finite__existsD,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => ? [X_12: A] : aa(A,$o,P2,X_12) ) ).

% not_finite_existsD
tff(fact_1305_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),modulo_modulo(A,A2,B2))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1306_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),modulo_modulo(A,A2,B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2) ) ) ) ).

% dvd_mod_iff
tff(fact_1307_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) ) ) ) ).

% dvd_trans
tff(fact_1308_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),A2) ) ).

% dvd_refl
tff(fact_1309_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X4: A,Xa2: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2),R)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_1310_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_an(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_1311_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ao(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_1312_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa),Xa,X) ) ).

% max_def_raw
tff(fact_1313_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ap(nat,fun(nat,$o),Mb))) ) ).

% finite_divisors_nat
tff(fact_1314_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),S2))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S2) ) ).

% pred_subset_eq2
tff(fact_1315_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1316_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_1317_finite__M__bounded__by__nat,axiom,
    ! [P2: fun(nat,$o),I2: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ar(fun(nat,$o),fun(nat,fun(nat,$o)),P2),I2))) ).

% finite_M_bounded_by_nat
tff(fact_1318_finite__less__ub,axiom,
    ! [F3: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),aa(nat,nat,F3,N2))
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_as(fun(nat,nat),fun(nat,fun(nat,$o)),F3),U))) ) ).

% finite_less_ub
tff(fact_1319_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,Mb,Nb))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1320_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C3)),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) ) ).

% mod_add_eq
tff(fact_1321_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,A3: A,B2: A,B3: A] :
          ( ( modulo_modulo(A,A2,C3) = modulo_modulo(A,A3,C3) )
         => ( ( modulo_modulo(A,B2,C3) = modulo_modulo(A,B3,C3) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),C3) ) ) ) ) ).

% mod_add_cong
tff(fact_1322_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C3)),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) ) ).

% mod_add_left_eq
tff(fact_1323_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) ) ).

% mod_add_right_eq
tff(fact_1324_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_1325_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),A2)
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1326_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) ) ).

% dvd_triv_right
tff(fact_1327_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C3)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3) ) ) ).

% dvd_mult_right
tff(fact_1328_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ).

% mult_dvd_mono
tff(fact_1329_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% dvd_triv_left
tff(fact_1330_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C3)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) ) ) ).

% dvd_mult_left
tff(fact_1331_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ).

% dvd_mult2
tff(fact_1332_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ).

% dvd_mult
tff(fact_1333_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
        <=> ? [K3: A] : A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvd_def
tff(fact_1334_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A2: A,B2: A,K2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% dvdI
tff(fact_1335_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ~ ! [K: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) ) ) ).

% dvdE
tff(fact_1336_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1337_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% unit_imp_dvd
tff(fact_1338_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),one_one(A)),A2) ) ).

% one_dvd
tff(fact_1339_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ) ) ).

% dvd_add
tff(fact_1340_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_1341_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) ) ) ) ).

% dvd_add_right_iff
tff(fact_1342_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( divide_divide(A,divide_divide(A,A2,D3),divide_divide(A,B2,D3)) = divide_divide(A,A2,B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1343_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( divide_divide(A,A2,C3) = divide_divide(A,B2,C3) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1344_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
           => ( ( divide_divide(A,A2,C3) = divide_divide(A,B2,C3) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1345_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,Mb,Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_1346_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_1347_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_1348_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_1349_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),zero_zero(nat))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_1350_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_1351_zdvd__antisym__nonneg,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Mb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
       => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Mb),Nb)
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Nb),Mb)
           => ( Mb = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_1352_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_at(nat,fun(nat,$o),Xb)) ).

% set_decode_def
tff(fact_1353_times__int__code_I1_J,axiom,
    ! [K2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K2),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_1354_times__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),times_times(int),zero_zero(int)),L) = zero_zero(int) ).

% times_int_code(2)
tff(fact_1355_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] :
          aa(num,A,numeral_numeral(A),bit0(Nb)) = $let(
            m2: A,
            m2:= aa(num,A,numeral_numeral(A),Nb),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),m2),m2) ) ) ).

% numeral_code(2)
tff(fact_1356_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_1357_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),Mb),Nb))
    <=> ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Mb)
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_1358_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1359_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(Mb)),nat_set_decode(Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_1360_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),modulo_modulo(A,A2,B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1361_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A2,B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1362_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_1363_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( modulo_modulo(A,A2,C3) = modulo_modulo(A,B2,C3) )
         => ~ ! [D2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D2)) ) ) ).

% mod_eqE
tff(fact_1364_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3))),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C3)),modulo_modulo(A,B2,C3)),C3)) ) ).

% div_add1_eq
tff(fact_1365_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),one_one(A)) ) ).

% not_is_unit_0
tff(fact_1366_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S)) ) ) ) ).

% minf(10)
tff(fact_1367_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S)) ) ) ) ).

% minf(9)
tff(fact_1368_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S)) ) ) ) ).

% pinf(10)
tff(fact_1369_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S)) ) ) ) ).

% pinf(9)
tff(fact_1370_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1371_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_1372_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_1373_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C3)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_1374_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1375_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C3)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1376_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_1377_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_1378_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) = $ite(aa(nat,nat,suc,modulo_modulo(nat,Mb,Nb)) = Nb,zero_zero(nat),aa(nat,nat,suc,modulo_modulo(nat,Mb,Nb))) ).

% mod_Suc
tff(fact_1379_finite__set__decode,axiom,
    ! [Nb: nat] : aa(set(nat),$o,finite_finite2(nat),nat_set_decode(Nb)) ).

% finite_set_decode
tff(fact_1380_mod__induct,axiom,
    ! [P2: fun(nat,$o),Nb: nat,P: nat,Mb: nat] :
      ( aa(nat,$o,P2,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),P)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),P)
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),P)
               => ( aa(nat,$o,P2,N2)
                 => aa(nat,$o,P2,modulo_modulo(nat,aa(nat,nat,suc,N2),P)) ) )
           => aa(nat,$o,P2,Mb) ) ) ) ) ).

% mod_induct
tff(fact_1381_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),C3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),divide_divide(A,C3,D3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1382_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),divide_divide(A,B2,C3)) ) ) ).

% dvd_mult_imp_div
tff(fact_1383_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)),A2)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A2,B2),C3) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_1384_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( divide_divide(A,A2,divide_divide(A,B2,C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C3) ) ) ) ) ).

% div_div_eq_right
tff(fact_1385_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ) ) ).

% div_mult_swap
tff(fact_1386_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C3)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C3) ) ) ) ).

% dvd_div_mult
tff(fact_1387_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),modulo_modulo(nat,Mb,Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_1388_gcd__nat__induct,axiom,
    ! [P2: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,M),zero_zero(nat))
     => ( ! [M: nat,N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P2,N2),modulo_modulo(nat,M,N2))
             => aa(nat,$o,aa(nat,fun(nat,$o),P2,M),N2) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P2,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_1389_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( ( divide_divide(A,B2,A2) = divide_divide(A,C3,A2) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_div_cancel
tff(fact_1390_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,A2,B2)),C3)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1391_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),divide_divide(A,C3,B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C3) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1392_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1393_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1394_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,Mb,aa(nat,nat,suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_1395_mod__eq__0D,axiom,
    ! [Mb: nat,D3: nat] :
      ( ( modulo_modulo(nat,Mb,D3) = zero_zero(nat) )
     => ? [Q3: nat] : Mb = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D3),Q3) ) ).

% mod_eq_0D
tff(fact_1396_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Yb: A,Nb: nat,Mb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Yb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,Xb),Nb)),aa(nat,A,power_power(A,Yb),Mb)) ) ) ) ).

% dvd_power_le
tff(fact_1397_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat,B2: A,Mb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A2),Nb)),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A2),Mb)),B2) ) ) ) ).

% power_le_dvd
tff(fact_1398_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Mb: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A2),Mb)),aa(nat,A,power_power(A,A2),Nb)) ) ) ).

% le_imp_power_dvd
tff(fact_1399_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Yb,Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Yb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1400_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_1401_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb) ) ) ).

% dvd_pos_nat
tff(fact_1402_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),Nb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_1403_zdvd__not__zless,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Mb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Mb),Nb)
       => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Nb),Mb) ) ) ).

% zdvd_not_zless
tff(fact_1404_bezout__lemma__nat,axiom,
    ! [D3: nat,A2: nat,B2: nat,Xb: nat,Yb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Yb)),D3) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Yb)),D3) ) )
         => ? [X3: nat,Y: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
              & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Y)),D3) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D3) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1405_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X3: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),A2)
      & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D2) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_1406_q__pos__lemma,axiom,
    ! [B3: int,Q6: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_1407_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B3: 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),Q2)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B3)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1408_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B3: 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),Q2)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q6),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1409_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),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),Q2)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q6),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1410_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),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),Q2)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1411_incr__mult__lemma,axiom,
    ! [D3: int,P2: fun(int,$o),K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X3: int] :
            ( aa(int,$o,P2,X3)
           => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
         => ! [X: int] :
              ( aa(int,$o,P2,X)
             => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1412_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1413_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Mb)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb) = one_one(int) )
      <=> ( ( Mb = one_one(int) )
          & ( Nb = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1414_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_1415_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
             => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_1416_finite__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_au(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) ) ).

% finite_lists_length_eq
tff(fact_1417_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,Mb,A2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & ( Mb != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_1418_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,Mb,A2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | ( Mb = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_1419_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        $let(
          half: nat,
          half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_1420_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1421_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A2,B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1422_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Nb)),aa(num,A,numeral_numeral(A),bit0(Q2))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1423_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_1424_finite__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_av(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) ) ).

% finite_lists_length_le
tff(fact_1425_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C3))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C3)),C3)) ) ).

% div_mult1_eq
tff(fact_1426_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2))),modulo_modulo(A,A2,B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_1427_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_1428_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_1429_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_1430_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2)) ) ).

% mod_div_decomp
tff(fact_1431_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ).

% cancel_div_mod_rules(1)
tff(fact_1432_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2))),modulo_modulo(A,A2,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ).

% cancel_div_mod_rules(2)
tff(fact_1433_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ).

% unit_dvdE
tff(fact_1434_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P2: fun(A,$o),L: A] :
          ( ? [X4: A] : aa(A,$o,P2,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4))
        <=> ? [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A)))
              & aa(A,$o,P2,X4) ) ) ) ).

% unity_coeff_ex
tff(fact_1435_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C3 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),D3)
               => ( ( divide_divide(A,B2,A2) = divide_divide(A,D3,C3) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D3) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1436_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),divide_divide(A,B2,C3))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1437_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,A2,B2)),C3)
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1438_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
           => ( ( divide_divide(A,B2,A2) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1439_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1440_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( ( divide_divide(A,A2,B2) = C3 )
          <=> ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_1441_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( ( A2 = divide_divide(A,C3,B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C3 ) ) ) ) ).

% unit_eq_div2
tff(fact_1442_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A2,B2),C3) ) ) ) ) ).

% div_mult_unit2
tff(fact_1443_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),B2) ) ) ) ).

% unit_div_commute
tff(fact_1444_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ) ) ).

% unit_div_mult_swap
tff(fact_1445_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),one_one(A))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A2,B2),C3) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1446_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,Mb,Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_1447_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A2),Nb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1448_div__less__mono,axiom,
    ! [A5: nat,B5: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A5),B5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( modulo_modulo(nat,A5,Nb) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B5,Nb) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,A5,Nb)),divide_divide(nat,B5,Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_1449_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ~ ! [S3: nat] : Mb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_1450_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ~ ! [S3: nat] : Nb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_1451_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Yb,Nb) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Yb),Xb)
       => ? [Q3: nat] : Xb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Yb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1452_dvd__imp__le,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb) ) ) ).

% dvd_imp_le
tff(fact_1453_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : modulo_modulo(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),modulo_modulo(nat,divide_divide(nat,Mb,Nb),Q2))),modulo_modulo(nat,Mb,Nb)) ).

% mod_mult2_eq
tff(fact_1454_div__mod__decomp,axiom,
    ! [A5: nat,Nb: nat] : A5 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,A5,Nb)),Nb)),modulo_modulo(nat,A5,Nb)) ).

% div_mod_decomp
tff(fact_1455_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1456_dvd__mult__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_1457_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D2: nat,X3: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),A2)
          & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1458_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( divide_divide(int,A2,B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1459_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( divide_divide(int,A2,B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1460_split__zdiv,axiom,
    ! [P2: fun(int,$o),Nb: int,K2: int] :
      ( aa(int,$o,P2,divide_divide(int,Nb,K2))
    <=> ( ( ( K2 = zero_zero(int) )
         => aa(int,$o,P2,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K2)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I4)),J3) ) )
             => aa(int,$o,P2,I4) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I4)),J3) ) )
             => aa(int,$o,P2,I4) ) ) ) ) ).

% split_zdiv
tff(fact_1461_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Yb )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Yb != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Yb != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Yb != $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                      $let(
                        half: nat,
                        half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_1462_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A)) ) ).

% even_zero
tff(fact_1463_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),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_1464_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),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_1465_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( aa(A,$o,aa(A,fun(A,$o),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,C3,A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C3),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1466_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% odd_even_add
tff(fact_1467_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,Mb: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,Xb),Mb)),aa(nat,A,power_power(A,Xb),Nb))
          <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ) ).

% dvd_power_iff
tff(fact_1468_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat,Xb: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
            | ( Xb = one_one(A) ) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),aa(nat,A,power_power(A,Xb),Nb)) ) ) ).

% dvd_power
tff(fact_1469_split__mod,axiom,
    ! [P2: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P2,modulo_modulo(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P2,Mb) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I4)),J3) )
               => aa(nat,$o,P2,J3) ) ) ) ) ) ).

% split_mod
tff(fact_1470_div2__even__ext__nat,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),bit0(one2))) = divide_divide(nat,Yb,aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xb)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Yb) )
       => ( Xb = Yb ) ) ) ).

% div2_even_ext_nat
tff(fact_1471_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),Mb)
      <=> ( Nb = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_1472_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Mb)),Mb)
      <=> ( Nb = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_1473_power__dvd__imp__le,axiom,
    ! [I2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,power_power(nat,I2),Mb)),aa(nat,nat,power_power(nat,I2),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_1474_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_1475_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_1476_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = 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),C3))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1477_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,B2),Nb)) ) ) ) ).

% power_mono_odd
tff(fact_1478_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),Mb) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1479_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% odd_pos
tff(fact_1480_dvd__power__iff__le,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,power_power(nat,K2),Mb)),aa(nat,nat,power_power(nat,K2),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_1481_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),B2)) = modulo_modulo(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1482_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),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),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_1483_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeLista: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeLista,S),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_1484_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat,Mb: nat] : modulo_modulo(A,divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)) = divide_divide(A,modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% div_exp_mod_exp_eq
tff(fact_1485_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ~ ! [B4: A] : A2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B4)),one_one(A)) ) ) ).

% oddE
tff(fact_1486_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Nb)) ) ) ).

% zero_le_even_power
tff(fact_1487_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_1488_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Nb))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1489_verit__le__mono__div,axiom,
    ! [A5: nat,B5: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A5),B5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,A5,Nb)),
                $ite(modulo_modulo(nat,B5,Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            divide_divide(nat,B5,Nb)) ) ) ).

% verit_le_mono_div
tff(fact_1490_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Vd2: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeLista,Vd2),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_1491_even__set__encode__iff,axiom,
    ! [A5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A5))
      <=> ~ member(nat,zero_zero(nat),A5) ) ) ).

% even_set_encode_iff
tff(fact_1492_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = divide_divide(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1493_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_1494_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,V),TreeLista,Vc),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_1495_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Nb))
        <=> ( ( Nb = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1496_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Yb) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Yb)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => (Yb) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)
               => ( (Yb)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_1497_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [S3: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_1498_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_1499_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( ( modulo_modulo(A,Xb,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Mb)) = modulo_modulo(A,Xb,Mb) )
              | ( modulo_modulo(A,Xb,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Mb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,Xb,Mb)),Mb) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1500_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [Vd: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_1501_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Nb)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1502_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_1503_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Yb) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Yb) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => (Yb) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va3,Vb2)
               => ( (Yb)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2)
                 => ( (Yb)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd)
                   => ( (Yb)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_1504_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va3,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_1505_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Xb,Mia)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Maa))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeLista,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_1506_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Yb) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Yb)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => (Yb) )
         => ( ( ? [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
             => (Yb) )
           => ( ( ? [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => (Yb) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
                   => ( (Yb)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_1507_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
           => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_1508_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))),one_one(A)) = divide_divide(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1509_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb) ) ) ) ).

% mult_less_iff1
tff(fact_1510_finite__nth__roots,axiom,
    ! [Nb: nat,C3: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(set(complex),$o,finite_finite2(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_aw(nat,fun(complex,fun(complex,$o)),Nb),C3))) ) ).

% finite_nth_roots
tff(fact_1511_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ax(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_1512_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Yb )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( Yb = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B4)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),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))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S3) )
               => ( ( Yb = vEBT_Node(Info2,zero_zero(nat),Ts2,S3) )
                 => ~ 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,S3)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3) )
                 => ( ( Yb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S3) )
                   => ~ 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,S3)),Xaa)) ) )
             => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) )
                   => ( ( Yb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( Yb = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                      aa(nat,product_prod(nat,nat),
                                        aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Xaa,Mi)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1513_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,Nb,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_1514_product__nth,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),Nb) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),divide_divide(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_1515_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),Xb: fun(A,B),Yb: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(A,$o)),I6),Xb)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(A,$o)),I6),Yb)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_az(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I6),Xb),Yb))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_1516_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),Xb: fun(A,B),Yb: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(A,$o)),I6),Xb)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(A,$o)),I6),Yb)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I6),Xb),Yb))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_1517_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Yb) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( (Yb)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ (Yb)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ~ (Yb)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ (Yb)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( (Yb)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1518_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_1519_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Nb,K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_1520_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K2)
       => ( modulo_modulo(int,K2,L) = K2 ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1521_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),L)
       => ( modulo_modulo(int,K2,L) = K2 ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1522_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_1523_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_1524_bot2E,axiom,
    ! [A: $tType,B: $tType,Xb: A,Yb: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),Xb),Yb) ).

% bot2E
tff(fact_1525_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,K2,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1526_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),modulo_modulo(int,K2,L)) ) ).

% neg_mod_bound
tff(fact_1527_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,K2,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_1528_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K2,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1529_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( modulo_modulo(int,I2,K2) = I2 )
    <=> ( ( K2 = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K2) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),I2) ) ) ) ).

% zmod_trivial_iff
tff(fact_1530_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A2,B2))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,A2,B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_1531_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,A2,B2)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),modulo_modulo(int,A2,B2)) ) ) ).

% neg_mod_conj
tff(fact_1532_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K2,L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K2)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_1533_zdiv__mono__strict,axiom,
    ! [A5: int,B5: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A5),B5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => ( ( modulo_modulo(int,A5,Nb) = zero_zero(int) )
         => ( ( modulo_modulo(int,B5,Nb) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A5,Nb)),divide_divide(int,B5,Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1534_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)),zero_zero(int))
       => ( modulo_modulo(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1535_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1536_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1537_split__zmod,axiom,
    ! [P2: fun(int,$o),Nb: int,K2: int] :
      ( aa(int,$o,P2,modulo_modulo(int,Nb,K2))
    <=> ( ( ( K2 = zero_zero(int) )
         => aa(int,$o,P2,Nb) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K2)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I4)),J3) ) )
             => aa(int,$o,P2,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I4)),J3) ) )
             => aa(int,$o,P2,J3) ) ) ) ) ).

% split_zmod
tff(fact_1538_split__neg__lemma,axiom,
    ! [K2: int,P2: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P2,divide_divide(int,Nb,K2)),modulo_modulo(int,Nb,K2))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P2,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1539_split__pos__lemma,axiom,
    ! [K2: int,P2: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),P2,divide_divide(int,Nb,K2)),modulo_modulo(int,Nb,K2))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K2)
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P2,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1540_verit__le__mono__div__int,axiom,
    ! [A5: int,B5: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A5),B5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A5,Nb)),
                $ite(modulo_modulo(int,B5,Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            divide_divide(int,B5,Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1541_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,Mb,A2))
        <=> ~ ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ( Mb = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_1542_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_1543_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_1544_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_1545_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Yb) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( (Yb)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ (Yb)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3) )
                 => ( ( (Yb)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1546_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),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),Mi),Ma)),zero_zero(nat),Va3,Vb2)),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2)),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_1547_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Yb) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Yb)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (Yb)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Yb)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va3,Vb2)),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2) )
                   => ( ( (Yb)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2)),Xaa)) ) )
               => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd) )
                     => ( ( (Yb)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_1548_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Mi: nat,Ma: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),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),Mi),Ma)),zero_zero(nat),Va3,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_1549_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1550_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_1551_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arsinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_1552_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Yb )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Yb = vEBT_Leaf($false,$false) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Yb = vEBT_Leaf($false,$false) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( Yb = $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                          $let(
                            half: nat,
                            half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_1553_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_1554_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_1555_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
            | ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1556_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : size_num(bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_1557_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A5: fun(B,A),B5: fun(B,A),Xb: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),minus_minus(fun(B,A)),A5),B5),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A5,Xb)),aa(B,A,B5,Xb)) ) ).

% minus_apply
tff(fact_1558_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_1559_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_1560_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_1561_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_1562_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_1563_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_1564_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C3: 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),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_1565_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_1566_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C3: 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),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_1567_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_1568_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_1569_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% diff_Suc_Suc
tff(fact_1570_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb)),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K2) ).

% Suc_diff_diff
tff(fact_1571_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Mb) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1572_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),Nb) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_1573_diff__diff__cancel,axiom,
    ! [I2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_1574_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) ).

% diff_diff_left
tff(fact_1575_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P2: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P2)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_1576_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = zero_zero(A) )
        <=> ~ (P2) ) ) ).

% of_bool_eq_0_iff
tff(fact_1577_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$false) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_1578_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P2: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( ~ (P2)
            & (Q) ) ) ) ).

% of_bool_less_iff
tff(fact_1579_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$true) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_1580_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = one_one(A) )
        <=> (P2) ) ) ).

% of_bool_eq_1_iff
tff(fact_1581_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_1582_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P2)
            | (Q) )) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_or_iff
tff(fact_1583_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1584_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1585_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1586_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1587_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_1588_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_1589_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ) ).

% div_diff
tff(fact_1590_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> (P2) ) ) ).

% zero_less_of_bool_iff
tff(fact_1591_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% zero_less_diff
tff(fact_1592_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1593_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_1594_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),one_one(A))
        <=> ~ (P2) ) ) ).

% of_bool_less_one_iff
tff(fact_1595_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P2: $o] : aa($o,A,zero_neq_one_of_bool(A),~ (P2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P2))) ) ).

% of_bool_not_iff
tff(fact_1596_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1597_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1598_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_1599_diff__Suc__1,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Nb)),one_one(nat)) = Nb ).

% diff_Suc_1
tff(fact_1600_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa($o,nat,zero_neq_one_of_bool(nat),Nb != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_1601_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))) = Nb ) ) ).

% Suc_pred
tff(fact_1602_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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),K2))),I2) = 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),K2),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1603_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1604_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Nb ) ) ).

% Suc_diff_1
tff(fact_1605_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ).

% even_diff
tff(fact_1606_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : divide_divide(A,aa($o,A,zero_neq_one_of_bool(A),(B2)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_1607_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_1608_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_1609_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1610_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : divide_divide(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_1611_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] : divide_divide(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_1612_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : modulo_modulo(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% one_mod_2_pow_eq
tff(fact_1613_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o,Q2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = aa($o,A,zero_neq_one_of_bool(A),(Q2)) )
        <=> ( (P)
          <=> (Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_1614_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A5: fun(A,B),B5: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A5),B5),X) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A5,X)),aa(A,B,B5,X)) ) ).

% fun_diff_def
tff(fact_1615_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A,D3: 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),C3),D3) )
         => ( ( A2 = B2 )
          <=> ( C3 = D3 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1616_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C3: 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),C3)),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)),C3) ) ).

% diff_right_commute
tff(fact_1617_diff__commute,axiom,
    ! [I2: nat,J: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),K2)),J) ).

% diff_commute
tff(fact_1618_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P2: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P2)
            & (Q) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_conj
tff(fact_1619_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1620_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) ) ) ).

% diff_right_mono
tff(fact_1621_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ) ).

% diff_left_mono
tff(fact_1622_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ) ).

% diff_mono
tff(fact_1623_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_1624_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) ) ) ).

% diff_strict_right_mono
tff(fact_1625_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_1626_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3) ) ) ) ).

% diff_eq_diff_less
tff(fact_1627_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ) ).

% diff_strict_mono
tff(fact_1628_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% left_diff_distrib
tff(fact_1629_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ).

% right_diff_distrib
tff(fact_1630_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C3: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)) ) ).

% left_diff_distrib'
tff(fact_1631_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ).

% right_diff_distrib'
tff(fact_1632_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% diff_diff_eq
tff(fact_1633_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) = A2 )
         => ( C3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_1634_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1635_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ).

% diff_add_eq
tff(fact_1636_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ).

% diff_diff_eq2
tff(fact_1637_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ).

% add_diff_eq
tff(fact_1638_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C3 ) ) ) ).

% eq_diff_eq
tff(fact_1639_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = C3 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) ) ) ) ).

% diff_eq_eq
tff(fact_1640_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A5: A,K2: A,A2: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A5),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1641_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C3: A,B2: A,D3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3)) ) ).

% add_diff_add
tff(fact_1642_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ).

% diff_divide_distrib
tff(fact_1643_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Z)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),Yb),Z)) ) ) ) ).

% dvd_diff
tff(fact_1644_zero__induct__lemma,axiom,
    ! [P2: fun(nat,$o),K2: nat,I2: nat] :
      ( aa(nat,$o,P2,K2)
     => ( ! [N2: nat] :
            ( aa(nat,$o,P2,aa(nat,nat,suc,N2))
           => aa(nat,$o,P2,N2) )
       => aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),I2)) ) ) ).

% zero_induct_lemma
tff(fact_1645_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb) = zero_zero(nat) )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_1646_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),zero_zero(nat)) = Mb ).

% minus_nat.diff_0
tff(fact_1647_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Nb)),K2) ) ).

% less_imp_diff_less
tff(fact_1648_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_1649_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Mb)) ) ).

% diff_le_mono2
tff(fact_1650_le__diff__iff_H,axiom,
    ! [A2: nat,C3: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_1651_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_1652_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),L)) ) ).

% diff_le_mono
tff(fact_1653_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1654_le__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_1655_eq__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_1656_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_1657_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_1658_diff__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% diff_cancel2
tff(fact_1659_Nat_Odiff__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% Nat.diff_cancel
tff(fact_1660_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2)) ).

% diff_mult_distrib
tff(fact_1661_diff__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) ).

% diff_mult_distrib2
tff(fact_1662_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Yb),Z)) ) ).

% max_diff_distrib_left
tff(fact_1663_dvd__diff__nat,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_1664_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P2))) ) ).

% zero_less_eq_of_bool
tff(fact_1665_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_1666_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P)) = $ite((P),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_1667_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: fun(A,$o),P: $o] :
          ( aa(A,$o,P2,aa($o,A,zero_neq_one_of_bool(A),(P)))
        <=> ( ( (P)
             => aa(A,$o,P2,one_one(A)) )
            & ( ~ (P)
             => aa(A,$o,P2,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_1668_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: fun(A,$o),P: $o] :
          ( aa(A,$o,P2,aa($o,A,zero_neq_one_of_bool(A),(P)))
        <=> ~ ( ( (P)
                & ~ aa(A,$o,P2,one_one(A)) )
              | ( ~ (P)
                & ~ aa(A,$o,P2,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_1669_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_1670_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_1671_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1672_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1673_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),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),C3),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1674_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),C3)),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)),C3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1675_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1676_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),C3),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1677_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),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),C3),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1678_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1679_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A2)) ) ) ).

% le_add_diff
tff(fact_1680_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1681_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ) ).

% le_diff_eq
tff(fact_1682_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% diff_le_eq
tff(fact_1683_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K2: A,Nb: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K2)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1684_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K2: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K2)) ) ) ).

% add_le_imp_le_diff
tff(fact_1685_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ) ).

% less_diff_eq
tff(fact_1686_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% diff_less_eq
tff(fact_1687_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),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_1688_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Yb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)) ) ).

% square_diff_square_factored
tff(fact_1689_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( C3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E3)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_1690_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( 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)),C3) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_1691_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xb: A,Yb: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),Yb),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_1692_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,A2,B2))) ) ).

% dvd_minus_mod
tff(fact_1693_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),aa(nat,nat,suc,Mb)) ).

% diff_less_Suc
tff(fact_1694_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_1695_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_1696_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_1697_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C3),A2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C3)) ) ) ).

% diff_less_mono
tff(fact_1698_less__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_1699_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1700_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_1701_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J) ) ).

% less_diff_conv
tff(fact_1702_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2) = K2 )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1703_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1704_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1705_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1706_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)),I2)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)) ) ).

% le_diff_conv
tff(fact_1707_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_1708_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_1709_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_1710_dvd__diffD1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),Mb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_1711_dvd__diffD,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),Mb) ) ) ) ).

% dvd_diffD
tff(fact_1712_mod__geq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( modulo_modulo(nat,Mb,Nb) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb) ) ) ).

% mod_geq
tff(fact_1713_mod__if,axiom,
    ! [Mb: nat,Nb: nat] :
      modulo_modulo(nat,Mb,Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb),Mb,modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ).

% mod_if
tff(fact_1714_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( modulo_modulo(nat,Mb,Nb) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb) ) ) ).

% le_mod_geq
tff(fact_1715_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Mb) ).

% nat_minus_add_max
tff(fact_1716_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E3)),D3)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1717_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E3)),C3)),D3) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1718_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E3)),D3)) ) ) ).

% less_add_iff2
tff(fact_1719_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E3)),C3)),D3) ) ) ).

% less_add_iff1
tff(fact_1720_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),divide_divide(A,B2,Z)) = $ite(Z = zero_zero(A),A2,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_1721_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Yb: A,Z: A,Xb: A,W2: A] :
          ( ( Yb != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Xb,Yb)),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),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Yb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1722_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,Yb,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),Xb),Z)),Yb),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1723_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Xb,Z)),Yb) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z)),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1724_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1725_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D5: A,Ta: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),D5)
         => ! [X: A,K4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Ta))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_1726_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D5: A,Ta: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),D5)
         => ! [X: A,K4: A] :
              ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Ta))
            <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_1727_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2)) = modulo_modulo(A,A2,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_1728_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_1729_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,A2,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_1730_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2))) = modulo_modulo(A,A2,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_1731_diff__Suc__less,axiom,
    ! [Nb: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,I2))),Nb) ) ).

% diff_Suc_less
tff(fact_1732_nat__diff__split__asm,axiom,
    ! [P2: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
            & ~ aa(nat,$o,P2,zero_zero(nat)) )
          | ? [D6: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
              & ~ aa(nat,$o,P2,D6) ) ) ) ).

% nat_diff_split_asm
tff(fact_1733_nat__diff__split,axiom,
    ! [P2: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
         => aa(nat,$o,P2,zero_zero(nat)) )
        & ! [D6: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
           => aa(nat,$o,P2,D6) ) ) ) ).

% nat_diff_split
tff(fact_1734_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)),I2)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)) ) ) ).

% less_diff_conv2
tff(fact_1735_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),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),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_1736_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( 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),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_1737_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_1738_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_1739_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb) )
      <=> ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1740_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_1741_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1742_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          divide_divide(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ))),
            aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))) ) ).

% exp_div_exp_eq
tff(fact_1743_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Yb: A,Z: A,Xb: A,W2: A] :
          ( ( Yb != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Yb)),divide_divide(A,W2,Z))
            <=> aa(A,$o,aa(A,fun(A,$o),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),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Yb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_1744_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Yb: A,Z: A,Xb: A,W2: A] :
          ( ( Yb != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Yb)),divide_divide(A,W2,Z))
            <=> aa(A,$o,aa(A,fun(A,$o),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),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Yb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_1745_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_1746_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Nb: nat,Mb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = divide_divide(A,aa(nat,A,power_power(A,A2),Mb),aa(nat,A,power_power(A,A2),Nb)) ) ) ) ) ).

% power_diff
tff(fact_1747_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1748_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_1749_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => ( divide_divide(nat,Mb,Nb) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ) ) ) ).

% div_geq
tff(fact_1750_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divide_divide(nat,Mb,Nb) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ) ).

% div_if
tff(fact_1751_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) = $ite(Mb = zero_zero(nat),Nb,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))),Nb))) ).

% add_eq_if
tff(fact_1752_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_1753_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_1754_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) = $ite(Mb = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))),Nb))) ).

% mult_eq_if
tff(fact_1755_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),Mb))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Q2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),Mb)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_1756_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Mb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),R2))
         => ( modulo_modulo(nat,Mb,Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_1757_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R2: A,S: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U)),S))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1758_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,Mb: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1759_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_1760_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)),K2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K2) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_1761_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,aa(nat,A,power_power(A,A2),Mb),aa(nat,A,power_power(A,A2),Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),divide_divide(A,one_one(A),aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1762_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P: A,Mb: nat] :
          aa(nat,A,power_power(A,P),Mb) = $ite(Mb = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),aa(nat,A,power_power(A,P),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1763_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),A2) = aa(nat,A,power_power(A,A2),Nb) ) ) ) ).

% power_minus_mult
tff(fact_1764_diff__le__diff__pow,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,K2),Mb)),aa(nat,nat,power_power(nat,K2),Nb))) ) ).

% diff_le_diff_pow
tff(fact_1765_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ( divide_divide(nat,Mb,Nb) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ) ) ) ).

% le_div_geq
tff(fact_1766_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_1767_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P2: fun(A,$o),A2: A] :
          ( ! [A4: A] :
              ( ( divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
             => aa(A,$o,P2,A4) )
         => ( ! [A4: A,B4: $o] :
                ( aa(A,$o,P2,A4)
               => ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4)),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
                 => aa(A,$o,P2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))) ) )
           => aa(A,$o,P2,A2) ) ) ) ).

% bits_induct
tff(fact_1768_int__power__div__base,axiom,
    ! [Mb: nat,K2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
       => ( divide_divide(int,aa(nat,int,power_power(int,K2),Mb),K2) = aa(nat,int,power_power(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1769_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] : modulo_modulo(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)) ) ).

% exp_mod_exp
tff(fact_1770_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Yb: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)),Yb)) ) ).

% power2_diff
tff(fact_1771_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1772_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),B2))),B2) = modulo_modulo(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1773_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_1774_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_1775_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_1776_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),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),bit0(one2))),Q2)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(num,A,numeral_numeral(A),L))),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),bit0(one2))),Q2)),R2)) ) ).

% divmod_step_eq
tff(fact_1777_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(Ta)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),one_one(nat)))) ) ).

% inrange
tff(fact_1778_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A] : aa(A,A,artanh(A),Xb) = divide_divide(A,aa(A,A,ln_ln(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% artanh_def
tff(fact_1779_Suc__if__eq,axiom,
    ! [A: $tType,F3: fun(nat,A),H: fun(nat,A),G3: A,Nb: nat] :
      ( ! [N2: nat] : aa(nat,A,F3,aa(nat,nat,suc,N2)) = aa(nat,A,H,N2)
     => ( ( aa(nat,A,F3,zero_zero(nat)) = G3 )
       => ( aa(nat,A,F3,Nb) = $ite(Nb = zero_zero(nat),G3,aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_1780_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_1781_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ).

% diff_shunt_var
tff(fact_1782_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_1783_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
     => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_1784_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_1785_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_1786_uminus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( uminus(A)
     => ! [A5: fun(B,A),Xb: B] : aa(B,A,aa(fun(B,A),fun(B,A),uminus_uminus(fun(B,A)),A5),Xb) = aa(A,A,uminus_uminus(A),aa(B,A,A5,Xb)) ) ).

% uminus_apply
tff(fact_1787_Diff__empty,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),bot_bot(set(A))) = A5 ).

% Diff_empty
tff(fact_1788_empty__Diff,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A5) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1789_Diff__cancel,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),A5) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1790_finite__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ).

% finite_Diff
tff(fact_1791_finite__Diff2,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
      <=> aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% finite_Diff2
tff(fact_1792_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Yb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ).

% compl_le_compl_iff
tff(fact_1793_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% neg_le_iff_le
tff(fact_1794_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_1795_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_1796_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_1797_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_1798_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_1799_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Yb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb) ) ) ).

% compl_less_compl_iff
tff(fact_1800_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% neg_less_iff_less
tff(fact_1801_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_right
tff(fact_1802_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) ) ).

% minus_mult_minus
tff(fact_1803_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_left
tff(fact_1804_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_1805_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_1806_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_1807_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_1808_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,uminus_uminus(A),Xb)),Yb)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Yb) ) ) ).

% minus_dvd_iff
tff(fact_1809_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),aa(A,A,uminus_uminus(A),Yb))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Yb) ) ) ).

% dvd_minus_iff
tff(fact_1810_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% Diff_eq_empty_iff
tff(fact_1811_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L2: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L2,H2) )
        <=> ( ( ( L = L2 )
              & ( H = H2 ) )
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
              & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L2),H2) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_1812_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( member(A,I2,set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_1813_ln__inj__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
       => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,ln_ln(real),Yb) )
        <=> ( Xb = Yb ) ) ) ) ).

% ln_inj_iff
tff(fact_1814_ln__less__cancel__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Yb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ) ) ).

% ln_less_cancel_iff
tff(fact_1815_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_1816_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or1337092689740270186AtMost(nat,L,U)) ).

% finite_atLeastAtMost
tff(fact_1817_semiring__norm_I80_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit1(Mb)),bit1(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ).

% semiring_norm(80)
tff(fact_1818_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_1819_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_le_0_iff_le
tff(fact_1820_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_1821_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_less_eq_nonneg
tff(fact_1822_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_1823_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_pos
tff(fact_1824_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_1825_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_0_iff_less
tff(fact_1826_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_1827_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_1828_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% verit_minus_simplify(3)
tff(fact_1829_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_1830_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb))) ) ).

% add_neg_numeral_simps(3)
tff(fact_1831_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_1832_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_1833_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] : divide_divide(A,Xb,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Xb) ) ).

% divide_minus1
tff(fact_1834_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_1835_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_1836_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1837_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_1838_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_1839_ln__le__cancel__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Yb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Yb) ) ) ) ).

% ln_le_cancel_iff
tff(fact_1840_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_1841_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_1842_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_1843_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( ( aa(real,real,ln_ln(real),Xb) = zero_zero(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_1844_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_1845_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ).

% zle_diff1_eq
tff(fact_1846_semiring__norm_I81_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit1(Mb)),bit0(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ).

% semiring_norm(81)
tff(fact_1847_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit1(Nb)) ).

% semiring_norm(77)
tff(fact_1848_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_1849_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_1850_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_1851_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_1852_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_1853_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_1854_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_1855_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),one_one(A)) = zero_zero(A) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_1856_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_1857_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_1858_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W2: num,Yb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W2)))),Yb) ) ).

% semiring_norm(168)
tff(fact_1859_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),Mb) ) ) ).

% neg_numeral_le_iff
tff(fact_1860_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Nb),Mb) ) ) ).

% neg_numeral_less_iff
tff(fact_1861_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ).

% take_bit_of_Suc_0
tff(fact_1862_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(Mb)),bit0(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_1863_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(Mb)),bit1(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_1864_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb)))
        <=> ( Mb != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_1865_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),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_1866_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_1867_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))) )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1868_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 )
        <=> $ite(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))),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1869_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( Mb != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_1870_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),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_1871_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_1872_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% take_bit_of_1
tff(fact_1873_add__neg__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_1874_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))
        <=> ( ( Nb = zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_1875_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),aa(num,nat,numeral_numeral(nat),V)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Mb),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_1876_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : divide_divide(nat,Mb,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = divide_divide(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_1877_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),aa(num,nat,numeral_numeral(nat),V)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Mb),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_1878_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : modulo_modulo(nat,Mb,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = modulo_modulo(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_1879_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),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_1880_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),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_1881_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: nat,Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% take_bit_of_exp
tff(fact_1882_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_1883_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_1884_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Yb)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Yb) ) ) ).

% compl_le_swap2
tff(fact_1885_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Yb)) ) ) ).

% compl_le_swap1
tff(fact_1886_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Yb)),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% compl_mono
tff(fact_1887_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Yb)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xb)),Yb) ) ) ).

% compl_less_swap2
tff(fact_1888_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,uminus_uminus(A),Yb)) ) ) ).

% compl_less_swap1
tff(fact_1889_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_1890_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_1891_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A5: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A5),X) = aa(B,B,uminus_uminus(B),aa(A,B,A5,X)) ) ).

% fun_Compl_def
tff(fact_1892_ln__div,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
       => ( aa(real,real,ln_ln(real),divide_divide(real,Xb,Yb)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Yb)) ) ) ) ).

% ln_div
tff(fact_1893_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ).

% take_bit_add
tff(fact_1894_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A,Mb: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_1895_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_1896_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Mb),Q2)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_1897_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_minus_iff
tff(fact_1898_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_le_iff
tff(fact_1899_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_imp_neg_le
tff(fact_1900_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% less_minus_iff
tff(fact_1901_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_less_iff
tff(fact_1902_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_1903_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_mult_commute
tff(fact_1904_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
        <=> ( ( A2 = B2 )
            | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% square_eq_iff
tff(fact_1905_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% is_num_normalize(8)
tff(fact_1906_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A5: A,K2: A,A2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A2) )
         => ( aa(A,A,uminus_uminus(A),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K2)),aa(A,A,uminus_uminus(A),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_1907_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_1908_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_1909_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_divide_left
tff(fact_1910_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ).

% minus_divide_divide
tff(fact_1911_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_1912_Diff__infinite__finite,axiom,
    ! [A: $tType,T2: set(A),S2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),T2)
     => ( ~ aa(set(A),$o,finite_finite2(A),S2)
       => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T2)) ) ) ).

% Diff_infinite_finite
tff(fact_1913_minus__int__code_I1_J,axiom,
    ! [K2: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),zero_zero(int)) = K2 ).

% minus_int_code(1)
tff(fact_1914_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_1915_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),Xb))),aa(real,real,uminus_uminus(real),Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_1916_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)),W2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W2)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W2)) ).

% int_distrib(3)
tff(fact_1917_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z22)) ).

% int_distrib(4)
tff(fact_1918_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ? [B4: A] : member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) ) ).

% psubset_imp_ex_mem
tff(fact_1919_zdvd__zdiffD,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),Nb))
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K2),Nb)
       => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K2),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_1920_ln__diff__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Yb))),divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Yb),Yb)) ) ) ).

% ln_diff_le
tff(fact_1921_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),one_one(real)) )
       => ( Xb = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_1922_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_1923_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_1924_take__bit__minus__small__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_1925_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_1926_ln__less__self,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_less_self
tff(fact_1927_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),K2)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_1928_not__take__bit__negative,axiom,
    ! [Nb: nat,K2: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_1929_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_1930_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),bit_se2584673776208193580ke_bit(A,Nb),bit_ri4674362597316999326ke_bit(A,Mb)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_1931_ex__nat__less,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( ? [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),Nb)
          & aa(nat,$o,P2,M2) )
    <=> ? [X4: nat] :
          ( member(nat,X4,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P2,X4) ) ) ).

% ex_nat_less
tff(fact_1932_all__nat__less,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),Nb)
         => aa(nat,$o,P2,M2) )
    <=> ! [X4: nat] :
          ( member(nat,X4,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P2,X4) ) ) ).

% all_nat_less
tff(fact_1933_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_le_neg_numeral
tff(fact_1934_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_le_numeral
tff(fact_1935_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_1936_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_neq_neg_numeral
tff(fact_1937_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_less_neg_numeral
tff(fact_1938_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_less_numeral
tff(fact_1939_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_1940_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_1941_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_1942_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_1943_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_1944_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_1945_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_1946_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_1947_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_1948_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_1949_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_1950_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_1951_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb) = one_one(A) )
        <=> ( ( Xb = one_one(A) )
            | ( Xb = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_1952_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B5: A,K2: A,B2: A,A2: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_1953_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_1954_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_1955_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se2638667681897837118et_bit(A,Mb,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se2638667681897837118et_bit(A,Mb,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_1956_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se5668285175392031749et_bit(A,Mb,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se5668285175392031749et_bit(A,Mb,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_1957_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se8732182000553998342ip_bit(A,Mb,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se8732182000553998342ip_bit(A,Mb,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_flip_bit_eq
tff(fact_1958_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) ) ) ) ).

% dvd_div_neg
tff(fact_1959_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) ) ) ) ).

% dvd_neg_div
tff(fact_1960_subset__Compl__self__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5))
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_1961_real__add__less__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Yb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_less_0_iff
tff(fact_1962_real__0__less__add__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Yb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),Xb)),Yb) ) ).

% real_0_less_add_iff
tff(fact_1963_int__le__induct,axiom,
    ! [I2: int,K2: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K2)
     => ( aa(int,$o,P2,K2)
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),K2)
             => ( aa(int,$o,P2,I3)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P2,I2) ) ) ) ).

% int_le_induct
tff(fact_1964_int__less__induct,axiom,
    ! [I2: int,K2: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K2)
     => ( aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),K2)
             => ( aa(int,$o,P2,I3)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P2,I2) ) ) ) ).

% int_less_induct
tff(fact_1965_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb) = one_one(int) )
     => ( ( Mb = one_one(int) )
        | ( Mb = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_1966_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb) = one_one(int) )
    <=> ( ( ( Mb = one_one(int) )
          & ( Nb = one_one(int) ) )
        | ( ( Mb = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( Nb = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_1967_ln__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_bound
tff(fact_1968_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_1969_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_1970_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_1971_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb)) ) ).

% ln_gt_zero
tff(fact_1972_ln__mult,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Yb)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Yb)) ) ) ) ).

% ln_mult
tff(fact_1973_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1974_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_1975_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_1976_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_le_neg_numeral
tff(fact_1977_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_1978_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_less_neg_numeral
tff(fact_1979_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_1980_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_1981_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),bit1(Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),aa(num,A,numeral_numeral(A),Nb))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_1982_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_1983_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_1984_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))) ) ).

% not_one_le_neg_numeral
tff(fact_1985_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_1986_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_1987_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Mb)) ) ).

% neg_one_le_numeral
tff(fact_1988_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_1989_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_1990_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))) ) ).

% not_one_less_neg_numeral
tff(fact_1991_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_1992_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Mb)) ) ).

% neg_one_less_numeral
tff(fact_1993_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_1994_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C3 = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_1995_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = C3 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_1996_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)) = A2 )
        <=> $ite(C3 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),A2 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_1997_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,uminus_uminus(A),B2),A2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_1998_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_1999_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),bit1(K2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),K2))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_2000_plusinfinity,axiom,
    ! [D3: int,P3: fun(int,$o),P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X3: int,K: int] :
            ( aa(int,$o,P3,X3)
          <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X3)
             => ( aa(int,$o,P2,X3)
              <=> aa(int,$o,P3,X3) ) )
         => ( ? [X_1: int] : aa(int,$o,P3,X_1)
           => ? [X_12: int] : aa(int,$o,P2,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_2001_minusinfinity,axiom,
    ! [D3: int,P1: fun(int,$o),P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X3: int,K: int] :
            ( aa(int,$o,P1,X3)
          <=> aa(int,$o,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Z3)
             => ( aa(int,$o,P2,X3)
              <=> aa(int,$o,P1,X3) ) )
         => ( ? [X_1: int] : aa(int,$o,P1,X_1)
           => ? [X_12: int] : aa(int,$o,P2,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_2002_int__induct,axiom,
    ! [P2: fun(int,$o),K2: int,I2: int] :
      ( aa(int,$o,P2,K2)
     => ( ! [I3: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),I3)
           => ( aa(int,$o,P2,I3)
             => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),K2)
             => ( aa(int,$o,P2,I3)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P2,I2) ) ) ) ).

% int_induct
tff(fact_2003_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] :
          aa(num,A,numeral_numeral(A),bit1(Nb)) = $let(
            m2: A,
            m2:= aa(num,A,numeral_numeral(A),Nb),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m2),m2)),one_one(A)) ) ) ).

% numeral_code(3)
tff(fact_2004_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
     => aa(set(nat),$o,finite_finite2(nat),N3) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_2005_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2006_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% take_bit_int_less_exp
tff(fact_2007_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2008_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2009_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2010_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2011_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2012_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2013_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,B2,C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2014_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,W2: num] :
          ( ( divide_divide(A,B2,C3) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
        <=> $ite(C3 != 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))),C3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2015_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit1(Nb)),aa(num,A,numeral_numeral(A),bit0(Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2016_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Yb: 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,Xb,Z))),Yb) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z)),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2017_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),B2) = $ite(Z = zero_zero(A),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_2018_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Yb: 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,Xb,Z))),Yb) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Z)),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2019_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),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_2020_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),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_2021_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),bit1(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_2022_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Nb) ).

% Suc3_eq_add_3
tff(fact_2023_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2024_decr__mult__lemma,axiom,
    ! [D3: int,P2: fun(int,$o),K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X3: int] :
            ( aa(int,$o,P2,X3)
           => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
         => ! [X: int] :
              ( aa(int,$o,P2,X)
             => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_2025_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K2)
       => ( modulo_modulo(int,K2,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_2026_verit__less__mono__div__int2,axiom,
    ! [A5: int,B5: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A5),B5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),Nb))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,B5,Nb)),divide_divide(int,A5,Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_2027_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_2028_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_2029_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_2030_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb) = Mb )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2031_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(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_2032_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_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_2033_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),K2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K2) ) ).

% take_bit_int_less_self_iff
tff(fact_2034_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2035_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2036_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2037_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2038_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2039_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2040_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2041_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2042_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),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))),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2043_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit1(Mb)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2044_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),bit1(Nb)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2045_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),Nb) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Mb),Nb) ).

% Suc_div_eq_add3_div
tff(fact_2046_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2)) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2047_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),Nb) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Mb),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_2048_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2049_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2) = K2 )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),K2)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2050_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),A2) ) ) ).

% take_bit_eq_0_iff
tff(fact_2051_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb)),Mb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_2052_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) = K2 )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2053_take__bit__int__eq__self,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_2054_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),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))),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2055_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2056_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_2057_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K2)
       => ( divide_divide(int,K2,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),K2),L),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_2058_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)),zero_zero(int))
       => ( divide_divide(int,K2,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_2059_take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) ) ).

% take_bit_int_greater_eq
tff(fact_2060_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),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,Nb),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_2061_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Nb)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2062_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2063_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_2064_ln__2__less__1,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(one2)))),one_one(real)) ).

% ln_2_less_1
tff(fact_2065_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          unique8689654367752047608divmod(A,bit0(Mb),bit1(Nb)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(Mb))),unique1321980374590559556d_step(A,bit1(Nb),unique8689654367752047608divmod(A,bit0(Mb),bit0(bit1(Nb))))) ) ).

% divmod_algorithm_code(7)
tff(fact_2066_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          unique8689654367752047608divmod(A,bit1(Mb),bit1(Nb)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit1(Mb))),unique1321980374590559556d_step(A,bit1(Nb),unique8689654367752047608divmod(A,bit1(Mb),bit0(bit1(Nb))))) ) ).

% divmod_algorithm_code(8)
tff(fact_2067_Bolzano,axiom,
    ! [A2: real,B2: real,P2: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A4: real,B4: real,C2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P2,A4),B4)
           => ( aa(real,$o,aa(real,fun(real,$o),P2,B4),C2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),B4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B4),C2)
                 => aa(real,$o,aa(real,fun(real,$o),P2,A4),C2) ) ) ) )
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
               => ? [D4: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                    & ! [A4: real,B4: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),X3)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B4)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B4),A4)),D4) )
                       => aa(real,$o,aa(real,fun(real,$o),P2,A4),B4) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P2,A2),B2) ) ) ) ).

% Bolzano
tff(fact_2068_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_bc(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_2069_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),$o,finite_finite2(int),set_or1337092689740270186AtMost(int,L,U)) ).

% finite_atLeastAtMost_int
tff(fact_2070_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: 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),F3),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),F3,A2),B2) ).

% case_prod_conv
tff(fact_2071_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num] : unique8689654367752047608divmod(A,Mb,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),Mb)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_2072_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,bit0(Nb)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_2073_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,bit1(Nb)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_2074_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : unique8689654367752047608divmod(A,bit0(Mb),bit0(Nb)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bd(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(5)
tff(fact_2075_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : unique8689654367752047608divmod(A,bit1(Mb),bit0(Nb)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_be(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(6)
tff(fact_2076_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,H: fun(B,A),F3: fun(C,fun(D,B)),Prod: product_prod(C,D)] : aa(B,A,H,aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),F3),Prod)) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_bf(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H),F3)),Prod) ).

% prod.case_distrib
tff(fact_2077_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),X1: B,X2: 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),F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X2)) = aa(C,A,aa(B,fun(C,A),F3,X1),X2) ).

% old.prod.case
tff(fact_2078_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q2: product_prod(A,B),F3: fun(A,fun(B,C)),G3: fun(A,fun(B,C)),P: product_prod(A,B)] :
      ( ! [X3: A,Y: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) = Q2 )
         => ( aa(B,C,aa(A,fun(B,C),F3,X3),Y) = aa(B,C,aa(A,fun(B,C),G3,X3),Y) ) )
     => ( ( P = Q2 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),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),G3),Q2) ) ) ) ).

% split_cong
tff(fact_2079_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P2: fun(B,fun(C,A)),Z: product_prod(B,C)] :
      ( aa(A,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P2),Z))
     => ~ ! [X3: B,Y: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P2,X3),Y)) ) ) ).

% case_prodE2
tff(fact_2080_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: 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_bg(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3)) = F3 ).

% case_prod_eta
tff(fact_2081_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),G3: fun(product_prod(A,B),C)] :
      ( ! [X3: A,Y: B] : aa(B,C,aa(A,fun(B,C),F3,X3),Y) = aa(product_prod(A,B),C,G3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) = G3 ) ) ).

% cond_case_prod_eta
tff(fact_2082_periodic__finite__ex,axiom,
    ! [D3: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X3: int,K: int] :
            ( aa(int,$o,P2,X3)
          <=> aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [X_13: int] : aa(int,$o,P2,X_13)
        <=> ? [X4: int] :
              ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D3))
              & aa(int,$o,P2,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2083_aset_I7_J,axiom,
    ! [D5: int,A5: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A5)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5)) ) ) ) ).

% aset(7)
tff(fact_2084_aset_I5_J,axiom,
    ! [D5: int,Ta: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,A5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_2085_aset_I4_J,axiom,
    ! [D5: int,Ta: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,A5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_2086_aset_I3_J,axiom,
    ! [D5: int,Ta: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_2087_bset_I7_J,axiom,
    ! [D5: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,B5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5)) ) ) ) ) ).

% bset(7)
tff(fact_2088_bset_I5_J,axiom,
    ! [D5: int,B5: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5)),Ta) ) ) ) ).

% bset(5)
tff(fact_2089_bset_I4_J,axiom,
    ! [D5: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,B5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X != Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_2090_bset_I3_J,axiom,
    ! [D5: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int)),B5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X = Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_2091_bset_I6_J,axiom,
    ! [D5: int,B5: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5)),Ta) ) ) ) ).

% bset(6)
tff(fact_2092_bset_I8_J,axiom,
    ! [D5: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int)),B5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5)) ) ) ) ) ).

% bset(8)
tff(fact_2093_aset_I6_J,axiom,
    ! [D5: int,Ta: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_2094_aset_I8_J,axiom,
    ! [D5: int,A5: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A5)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5)) ) ) ) ).

% aset(8)
tff(fact_2095_cpmi,axiom,
    ! [D5: int,P2: fun(int,$o),P3: fun(int,$o),B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Z3)
           => ( aa(int,$o,P2,X3)
            <=> aa(int,$o,P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,B5)
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,P2,X3)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5)) ) )
         => ( ! [X3: int,K: int] :
                ( aa(int,$o,P3,X3)
              <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D5))) )
           => ( ? [X_13: int] : aa(int,$o,P2,X_13)
            <=> ( ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P3,X4) )
                | ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa2: int] :
                        ( member(int,Xa2,B5)
                        & aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2096_cppi,axiom,
    ! [D5: int,P2: fun(int,$o),P3: fun(int,$o),A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X3)
           => ( aa(int,$o,P2,X3)
            <=> aa(int,$o,P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,A5)
                     => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,P2,X3)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)) ) )
         => ( ! [X3: int,K: int] :
                ( aa(int,$o,P3,X3)
              <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D5))) )
           => ( ? [X_13: int] : aa(int,$o,P2,X_13)
            <=> ( ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P3,X4) )
                | ? [X4: int] :
                    ( member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa2: int] :
                        ( member(int,Xa2,A5)
                        & aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa2),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2097_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : unique8689654367752047608divmod(A,Mb,Nb) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),divide_divide(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Nb))),modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Nb))) ) ).

% divmod_def
tff(fact_2098_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique8689654367752047608divmod(nat,Mb,Nb) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,aa(num,nat,numeral_numeral(nat),Mb),aa(num,nat,numeral_numeral(nat),Nb))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),Mb),aa(num,nat,numeral_numeral(nat),Nb))) ).

% divmod'_nat_def
tff(fact_2099_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_bh(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_2100_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          unique8689654367752047608divmod(A,Mb,Nb) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),Mb)),unique1321980374590559556d_step(A,Nb,unique8689654367752047608divmod(A,Mb,bit0(Nb)))) ) ).

% divmod_divmod_step
tff(fact_2101_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),Xb)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2102_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),Mb),
        aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bi(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_2103_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: int] :
          aa(int,A,ring_1_of_int(A),K2) = $ite(
            K2 = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)),
              aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K2))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2))))),
                $ite(modulo_modulo(int,K2,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_2104_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit1(K2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K2))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_2105_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,U,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),divide_divide(real,U,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Yb)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_2106_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2107_of__int__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W2: int,Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),W2) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( W2 = Z ) ) ) ).

% of_int_eq_iff
tff(fact_2108_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B),C3: fun(A,fun(B,$o))] :
      ( ! [A4: A,B4: B] :
          ( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => aa(B,$o,aa(A,fun(B,$o),C3,A4),B4) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C3),P) ) ).

% case_prodI2
tff(fact_2109_case__prodI,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F3,A2),B2)
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) ) ).

% case_prodI
tff(fact_2110_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: product_prod(A,B),Z: C,C3: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B4: B] :
          ( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C3,A4),B4)) )
     => 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)),C3),P)) ) ).

% mem_case_prodI2
tff(fact_2111_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C3: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C3,A2),B2))
     => member(A,Z,aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C3),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_2112_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: product_prod(A,B),C3: fun(A,fun(B,fun(C,$o))),Xb: 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) = P )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C3,A4),B4),Xb) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C3),P),Xb) ) ).

% case_prodI2'
tff(fact_2113_real__sqrt__less__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Yb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ).

% real_sqrt_less_iff
tff(fact_2114_of__int__of__bool,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P2: $o] : aa(int,A,ring_1_of_int(A),aa($o,int,zero_neq_one_of_bool(int),(P2))) = aa($o,A,zero_neq_one_of_bool(A),(P2)) ) ).

% of_int_of_bool
tff(fact_2115_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_2116_tanh__real__less__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xb)),aa(real,real,tanh(real),Yb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ).

% tanh_real_less_iff
tff(fact_2117_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_2118_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_2119_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_2120_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ) ).

% of_int_le_iff
tff(fact_2121_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,Nb: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( Z = aa(num,int,numeral_numeral(int),Nb) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_2122_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K2)) = aa(num,A,numeral_numeral(A),K2) ) ).

% of_int_numeral
tff(fact_2123_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% of_int_less_iff
tff(fact_2124_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = one_one(A) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_2125_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_2126_of__int__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_mult
tff(fact_2127_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_add
tff(fact_2128_of__int__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),Z)) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_minus
tff(fact_2129_real__sqrt__gt__0__iff,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Yb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb) ) ).

% real_sqrt_gt_0_iff
tff(fact_2130_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_2131_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),minus_minus(int),W2),Z)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_diff
tff(fact_2132_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_2133_real__sqrt__gt__1__iff,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,sqrt,Yb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Yb) ) ).

% real_sqrt_gt_1_iff
tff(fact_2134_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2135_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: int,B2: int,W2: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Xb) = aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2) )
        <=> ( Xb = aa(nat,int,power_power(int,B2),W2) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_2136_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W2: nat,Xb: int] :
          ( ( aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2) = aa(int,A,ring_1_of_int(A),Xb) )
        <=> ( aa(nat,int,power_power(int,B2),W2) = Xb ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_2137_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,Nb: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,power_power(int,Z),Nb)) = aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),Z)),Nb) ) ).

% of_int_power
tff(fact_2138_tanh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_2139_tanh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tanh(real),Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% tanh_real_pos_iff
tff(fact_2140_less__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),pred_numeral(K2)) ) ).

% less_Suc_numeral
tff(fact_2141_less__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K2)),Nb) ) ).

% less_numeral_Suc
tff(fact_2142_le__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),pred_numeral(K2)) ) ).

% le_Suc_numeral
tff(fact_2143_le__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K2)),Nb) ) ).

% le_numeral_Suc
tff(fact_2144_max__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),pred_numeral(K2))) ).

% max_Suc_numeral
tff(fact_2145_max__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K2)),Nb)) ).

% max_numeral_Suc
tff(fact_2146_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_2147_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_2148_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_2149_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_2150_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_2151_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_le_numeral_iff
tff(fact_2152_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_less_numeral_iff
tff(fact_2153_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_2154_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_2155_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_2156_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_2157_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_2158_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B2: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),aa(nat,int,power_power(int,B2),W2)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2159_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,B2),W2)),Xb) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2160_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Yb: int,Xb: num,Nb: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Yb) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xb)),Nb) )
        <=> ( Yb = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_2161_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: nat,Yb: int] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xb)),Nb) = aa(int,A,ring_1_of_int(A),Yb) )
        <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb) = Yb ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_2162_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,B2),W2)),Xb) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2163_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B2: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B2)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,power_power(int,B2),W2)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2164_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2165_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2166_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2167_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2168_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Yb: int,Xb: num,Nb: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Yb) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb) )
        <=> ( Yb = aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_2169_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: nat,Yb: int] :
          ( ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb) = aa(int,A,ring_1_of_int(A),Yb) )
        <=> ( aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb) = Yb ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2170_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2171_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2172_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2173_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_le_of_int
tff(fact_2174_real__sqrt__less__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Yb)) ) ).

% real_sqrt_less_mono
tff(fact_2175_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_less_of_int
tff(fact_2176_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),Xb) ) ).

% ex_of_int_less
tff(fact_2177_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int,Yb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Xb)),Yb) = aa(A,A,aa(A,fun(A,A),times_times(A),Yb),aa(int,A,ring_1_of_int(A),Xb)) ) ).

% mult_of_int_commute
tff(fact_2178_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C3: fun(B,fun(C,set(A))),P: product_prod(B,C)] :
      ( member(A,Z,aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C3),P))
     => ~ ! [X3: B,Y: C] :
            ( ( P = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y) )
           => ~ member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C3,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_2179_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,Yb: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(int,A,ring_1_of_int(A),Xb)),aa(int,A,ring_1_of_int(A),Yb)) ) ).

% of_int_max
tff(fact_2180_case__prodE,axiom,
    ! [A: $tType,B: $tType,C3: fun(A,fun(B,$o)),P: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C3),P)
     => ~ ! [X3: A,Y: B] :
            ( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C3,X3),Y) ) ) ).

% case_prodE
tff(fact_2181_case__prodD,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))
     => aa(B,$o,aa(A,fun(B,$o),F3,A2),B2) ) ).

% case_prodD
tff(fact_2182_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(A,fun(B,fun(C,$o))),P: product_prod(A,B),Z: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C3),P),Z)
     => ~ ! [X3: A,Y: B] :
            ( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C3,X3),Y),Z) ) ) ).

% case_prodE'
tff(fact_2183_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,$o))),A2: A,B2: B,C3: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C3)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R,A2),B2),C3) ) ).

% case_prodD'
tff(fact_2184_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_2185_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,fun(B,$o)),B5: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A5),B5)
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A5))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),B5))) ) ).

% Collect_case_prod_mono
tff(fact_2186_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xb)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_2187_tanh__real__gt__neg1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),Xb)) ).

% tanh_real_gt_neg1
tff(fact_2188_sqrt2__less__2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sqrt2_less_2
tff(fact_2189_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_2190_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_pos
tff(fact_2191_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [X3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X3)),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int))))
          & ! [Y3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y3)),Xb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y3),one_one(int)))) )
             => ( Y3 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_2192_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))) ) ) ).

% floor_exists
tff(fact_2193_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2)) ) ).

% of_int_neg_numeral
tff(fact_2194_int__le__real__less,axiom,
    ! [Nb: int,Mb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),Mb)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Nb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Mb)),one_one(real))) ) ).

% int_le_real_less
tff(fact_2195_int__less__real__le,axiom,
    ! [Nb: int,Mb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),Mb)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Nb)),one_one(real))),aa(int,real,ring_1_of_int(real),Mb)) ) ).

% int_less_real_le
tff(fact_2196_real__less__rsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Yb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,sqrt,Yb)) ) ).

% real_less_rsqrt
tff(fact_2197_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),U)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_2198_real__less__lsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Yb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,power_power(real,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),Yb) ) ) ) ).

% real_less_lsqrt
tff(fact_2199_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,Xb)) = divide_divide(real,aa(real,real,ln_ln(real),Xb),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_2200_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),divide_divide(nat,Mb,Nb)),modulo_modulo(nat,Mb,Nb)) ).

% divmod_nat_def
tff(fact_2201_arsinh__real__aux,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_2202_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Yb))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Yb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,Xb) = Yb ) ) ) ) ).

% round_unique
tff(fact_2203_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))) ) ).

% of_int_round_gt
tff(fact_2204_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))) ) ).

% of_int_round_ge
tff(fact_2205_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_2206_int__ge__less__than__def,axiom,
    ! [D3: int] : int_ge_less_than(D3) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_bj(int,fun(int,fun(int,$o)),D3))) ).

% int_ge_less_than_def
tff(fact_2207_int__ge__less__than2__def,axiom,
    ! [D3: int] : int_ge_less_than2(D3) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_bk(int,fun(int,fun(int,$o)),D3))) ).

% int_ge_less_than2_def
tff(fact_2208_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Mb),Nb)
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_bl(int,int)),set_or1337092689740270186AtMost(int,Mb,Nb)) = divide_divide(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),Mb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_2209_split__part,axiom,
    ! [B: $tType,A: $tType,P2: $o,Q: fun(A,fun(B,$o)),X: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bm($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P2)),Q)),X)
    <=> ( (P2)
        & aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q),X) ) ) ).

% split_part
tff(fact_2210_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_bn(B,A)),A5) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2211_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_2212_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F4: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),F4) = zero_zero(B) )
          <=> ! [X4: A] :
                ( member(A,X4,F4)
               => ( aa(A,B,F3,X4) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_2213_sum_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_2214_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2215_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( 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_bo(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_2216_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( 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_bp(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_2217_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F3: fun(B,int),A5: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_bq(fun(B,int),fun(B,A),F3)),A5) ) ).

% of_int_sum
tff(fact_2218_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_br(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_2219_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => ( aa(A,B,G3,X3) = zero_zero(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_2220_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A5: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A5)
               => ( aa(B,A,G3,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_2221_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,K5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),aa(A,B,G3,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),K5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),K5)) ) ) ).

% sum_mono
tff(fact_2222_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),H: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_bs(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A5) = 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),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),A5)) ) ).

% sum.distrib
tff(fact_2223_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A5: set(A),B5: set(B),G3: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(B),$o,finite_finite2(B),B5)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_bu(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B5),G3),R)),A5) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_bw(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A5),G3),R)),B5) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2224_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_2225_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)) ) ) ).

% sum_nonneg
tff(fact_2226_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F3: fun(B,A),I6: set(B),G3: fun(B,A),I2: B] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),I6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),I6) )
         => ( ! [I3: B] :
                ( member(B,I3,I6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,I3)),aa(B,A,G3,I3)) )
           => ( member(B,I2,I6)
             => ( aa(set(B),$o,finite_finite2(B),I6)
               => ( aa(B,A,F3,I2) = aa(B,A,G3,I2) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_2227_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(set(A),fun(fun(A,$o),fun(A,$o)),A5),P2))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_by(fun(A,B),fun(fun(A,$o),fun(A,B)),G3),P2)),A5) ) ) ) ).

% sum.inter_filter
tff(fact_2228_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5) = zero_zero(B) )
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => ( aa(A,B,F3,X4) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_2229_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S: set(A),Ta: set(B),G3: fun(B,C),I2: fun(B,A),F3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),Ta)
           => ( ! [X3: B] :
                  ( member(B,X3,Ta)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G3,X3)) )
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ? [Xa: B] :
                        ( member(B,Xa,Ta)
                        & ( aa(B,A,I2,Xa) = X3 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,X3)),aa(B,C,G3,Xa)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),F3),S)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G3),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_2230_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
           => ( ? [X: A] :
                  ( member(A,X,A5)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,G3,X)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_2231_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S2: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X15: A,Y15: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),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),X22),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S2)
             => ( ! [X3: B] :
                    ( member(B,X3,S2)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X3)),aa(B,A,G3,X3)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S2)) ) ) ) ) ) ).

% sum.related
tff(fact_2232_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A] :
                  ( member(A,X3,A5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5)) ) ) ) ) ).

% sum_strict_mono
tff(fact_2233_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T3: set(B),S2: set(A),I2: fun(B,A),J: fun(A,B),T2: set(B),G3: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5))
                 => ( aa(B,A,I2,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                     => ( aa(A,B,J,aa(B,A,I2,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                       => member(A,aa(B,A,I2,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S5)
                         => ( aa(A,C,G3,A4) = zero_zero(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T3)
                           => ( aa(B,C,H,B4) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S2)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G3,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),G3),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),T2) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_2234_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xb)),archimedean_round(A,Yb)) ) ) ).

% round_mono
tff(fact_2235_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F3: fun(A,B),B5: B,I2: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I3: A] :
                ( member(A,I3,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I3)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),S) = B5 )
             => ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),B5) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2236_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F3: fun(A,B),I2: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I3: A] :
                ( member(A,I3,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I3)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),S) = zero_zero(B) )
             => ( member(A,I2,S)
               => ( aa(A,B,F3,I2) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2237_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bz(fun(A,B),fun(A,$o),G3)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2238_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),I2: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( member(A,I2,I6)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,I2))
             => ( ! [I3: A] :
                    ( member(A,I3,I6)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),I6)) ) ) ) ) ) ).

% sum_pos2
tff(fact_2239_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ( I6 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I6)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,I3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),I6)) ) ) ) ) ).

% sum_pos
tff(fact_2240_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S2: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,G3,X3) = zero_zero(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),S2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_2241_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S2: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,H,X3) = zero_zero(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_2242_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,G3,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),S2) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_2243_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,G3,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),T2) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_2244_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5))
                   => ( aa(A,B,G3,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B5))
                     => ( aa(A,B,H,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C5) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B5) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_2245_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5))
                   => ( aa(A,B,G3,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B5))
                     => ( aa(A,B,H,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B5) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C5) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_2246_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B5: set(A),A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),B5)) ) ) ) ) ).

% sum.subset_diff
tff(fact_2247_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = 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),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B5)) ) ) ) ) ).

% sum_diff
tff(fact_2248_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,B4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B5)) ) ) ) ) ).

% sum_mono2
tff(fact_2249_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B5: set(A),A5: set(A),B2: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( member(A,B2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,B2))
               => ( ! [X3: A] :
                      ( member(A,X3,B5)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B5)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_2250_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(Nb)))) ) ).

% mask_numeral
tff(fact_2251_set__encode__insert,axiom,
    ! [A5: set(nat),Nb: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),A5)
     => ( ~ member(nat,Nb,A5)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Nb),A5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),aa(set(nat),nat,nat_set_encode,A5)) ) ) ) ).

% set_encode_insert
tff(fact_2252_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ca(nat,nat)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_2253_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),Xb) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2254_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2255_arith__series__nat,axiom,
    ! [A2: nat,D3: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_cb(nat,fun(nat,fun(nat,nat)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),D3))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_2256_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% mask_nat_positive_iff
tff(fact_2257_singletonI,axiom,
    ! [A: $tType,A2: A] : member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ).

% singletonI
tff(fact_2258_finite__insert,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5))
    <=> aa(set(A),$o,finite_finite2(A),A5) ) ).

% finite_insert
tff(fact_2259_singleton__conv,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cc(A,fun(A,$o),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_2260_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),fequal(A),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_2261_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A5: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2262_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A5: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2263_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)),insert(A),A2),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_2264_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C3),bot_bot(set(A))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C3 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_2265_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),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)),insert(A),A2),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) ).

% insert_Diff_single
tff(fact_2266_finite__Diff__insert,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] :
      ( aa(set(A),$o,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)),insert(A),A2),B5)))
    <=> aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ).

% finite_Diff_insert
tff(fact_2267_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_2268_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_2269_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( ( bit_se2239418461657761734s_mask(A,Nb) = zero_zero(A) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_2270_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),Xb: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5)) ) ) ) ) ).

% sum.insert
tff(fact_2271_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_2272_subset__Compl__singleton,axiom,
    ! [A: $tType,A5: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
    <=> ~ member(A,B2,A5) ) ).

% subset_Compl_singleton
tff(fact_2273_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),Z) ) ).

% ceiling_add_of_int
tff(fact_2274_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_2275_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb) ) ) ).

% zero_less_ceiling
tff(fact_2276_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(num,A,numeral_numeral(A),V)) ) ) ).

% ceiling_le_numeral
tff(fact_2277_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_2278_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb) ) ) ).

% one_le_ceiling
tff(fact_2279_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V)),Xb) ) ) ).

% numeral_less_ceiling
tff(fact_2280_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_2281_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ).

% one_less_ceiling
tff(fact_2282_set__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Nb,Xb)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_2283_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_2284_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_2285_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb)))) ) ).

% sum.cl_ivl_Suc
tff(fact_2286_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_2287_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) ) ) ).

% zero_le_ceiling
tff(fact_2288_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: fun(nat,A),A5: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cd(fun(nat,A),fun(nat,A),C3)),A5) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A5)
            & member(nat,zero_zero(nat),A5) ),
            aa(nat,A,C3,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_2289_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_2290_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),Xb) ) ) ).

% numeral_le_ceiling
tff(fact_2291_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2292_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xb) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2293_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: fun(nat,A),D3: fun(nat,A),A5: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_ce(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D3)),A5) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A5)
            & member(nat,zero_zero(nat),A5) ),
            divide_divide(A,aa(nat,A,C3,zero_zero(nat)),aa(nat,A,D3,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_2294_singletonD,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( member(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_2295_singleton__iff,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( member(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_2296_doubleton__eq__iff,axiom,
    ! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),D3),bot_bot(set(A)))) )
    <=> ( ( ( A2 = C3 )
          & ( B2 = D3 ) )
        | ( ( A2 = D3 )
          & ( B2 = C3 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2297_insert__not__empty,axiom,
    ! [A: $tType,A2: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_2298_singleton__inject,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_2299_finite_OinsertI,axiom,
    ! [A: $tType,A5: set(A),A2: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)) ) ).

% finite.insertI
tff(fact_2300_less__eq__mask,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),bit_se2239418461657761734s_mask(nat,Nb)) ).

% less_eq_mask
tff(fact_2301_sum__diff1__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A),A2: A] :
      aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),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)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),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),F3),A5)),aa(A,nat,F3,A2)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)) ).

% sum_diff1_nat
tff(fact_2302_Collect__conv__if,axiom,
    ! [A: $tType,A2: A,P2: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cf(A,fun(fun(A,$o),fun(A,$o)),A2),P2)) = $ite(aa(A,$o,P2,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_2303_Collect__conv__if2,axiom,
    ! [A: $tType,A2: A,P2: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(A,fun(fun(A,$o),fun(A,$o)),A2),P2)) = $ite(aa(A,$o,P2,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_2304_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(nat),F3: fun(nat,A),G3: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A5)
         => ( ! [X3: nat] :
                ( member(nat,aa(nat,nat,suc,X3),A5)
               => ( aa(nat,A,F3,aa(nat,nat,suc,X3)) = aa(nat,A,G3,aa(nat,nat,suc,X3)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),A5) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),A5) ) ) ) ) ).

% sum_cong_Suc
tff(fact_2305_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Yb)),archimedean_ceiling(A,Xb)) ) ) ).

% ceiling_mono
tff(fact_2306_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ).

% le_of_int_ceiling
tff(fact_2307_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Yb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb) ) ) ).

% ceiling_less_cancel
tff(fact_2308_infinite__finite__induct,axiom,
    ! [A: $tType,P2: fun(set(A),$o),A5: set(A)] :
      ( ! [A6: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A6)
         => aa(set(A),$o,P2,A6) )
     => ( aa(set(A),$o,P2,bot_bot(set(A)))
       => ( ! [X3: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ member(A,X3,F5)
               => ( aa(set(A),$o,P2,F5)
                 => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),F5)) ) ) )
         => aa(set(A),$o,P2,A5) ) ) ) ).

% infinite_finite_induct
tff(fact_2309_finite__ne__induct,axiom,
    ! [A: $tType,F4: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ( F4 != bot_bot(set(A)) )
       => ( ! [X3: A] : aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))
         => ( ! [X3: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ member(A,X3,F5)
                   => ( aa(set(A),$o,P2,F5)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),F5)) ) ) ) )
           => aa(set(A),$o,P2,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_2310_finite__induct,axiom,
    ! [A: $tType,F4: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,P2,bot_bot(set(A)))
       => ( ! [X3: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ member(A,X3,F5)
               => ( aa(set(A),$o,P2,F5)
                 => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),F5)) ) ) )
         => aa(set(A),$o,P2,F4) ) ) ) ).

% finite_induct
tff(fact_2311_finite_Osimps,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,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)),insert(A),A7),A8) )
            & aa(set(A),$o,finite_finite2(A),A8) ) ) ) ).

% finite.simps
tff(fact_2312_finite_Ocases,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( ( A2 != bot_bot(set(A)) )
       => ~ ! [A6: set(A)] :
              ( ? [A4: A] : A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),A6)
             => ~ aa(set(A),$o,finite_finite2(A),A6) ) ) ) ).

% finite.cases
tff(fact_2313_subset__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
    <=> ( ( X5 = bot_bot(set(A)) )
        | ( X5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_2314_subset__singletonD,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))
     => ( ( A5 = bot_bot(set(A)) )
        | ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_2315_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)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_2316_Diff__insert__absorb,axiom,
    ! [A: $tType,Xb: A,A5: set(A)] :
      ( ~ member(A,Xb,A5)
     => ( 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)),insert(A),Xb),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = A5 ) ) ).

% Diff_insert_absorb
tff(fact_2317_Diff__insert2,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(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)),insert(A),A2),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))),B5) ).

% Diff_insert2
tff(fact_2318_insert__Diff,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( member(A,A2,A5)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),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)),insert(A),A2),bot_bot(set(A))))) = A5 ) ) ).

% insert_Diff
tff(fact_2319_Diff__insert,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(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)),insert(A),A2),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_2320_sum__subtractf__nat,axiom,
    ! [A: $tType,A5: set(A),G3: fun(A,nat),F3: fun(A,nat)] :
      ( ! [X3: A] :
          ( member(A,X3,A5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G3,X3)),aa(A,nat,F3,X3)) )
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_ch(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G3),F3)),A5) = 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),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G3),A5)) ) ) ).

% sum_subtractf_nat
tff(fact_2321_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ci(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2322_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2239418461657761734s_mask(int,Nb)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_2323_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_cj(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2324_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X4: A] :
            ( member(A,X4,A5)
            & ( aa(A,nat,F3,X4) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa2: A] :
                ( member(A,Xa2,A5)
               => ( ( X4 != Xa2 )
                 => ( aa(A,nat,F3,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_2325_sum__SucD,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A),Nb: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5) = aa(nat,nat,suc,Nb) )
     => ? [X3: A] :
          ( member(A,X3,A5)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X3)) ) ) ).

% sum_SucD
tff(fact_2326_sum__eq__1__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5) = one_one(nat) )
      <=> ? [X4: A] :
            ( member(A,X4,A5)
            & ( aa(A,nat,F3,X4) = one_one(nat) )
            & ! [Xa2: A] :
                ( member(A,Xa2,A5)
               => ( ( X4 != Xa2 )
                 => ( aa(A,nat,F3,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2327_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),A2) ) ) ).

% ceiling_le
tff(fact_2328_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_2329_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% less_ceiling_iff
tff(fact_2330_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Yb))) ) ).

% ceiling_add_le
tff(fact_2331_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),P2: fun(set(A),$o),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),$o,P2,bot_bot(set(A)))
           => ( ! [X3: A,S6: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S6)
                 => ( ! [Y3: A] :
                        ( member(A,Y3,S6)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Y3)),aa(A,B,F3,X3)) )
                   => ( aa(set(A),$o,P2,S6)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),S6)) ) ) )
             => aa(set(A),$o,P2,S2) ) ) ) ) ).

% finite_ranking_induct
tff(fact_2332_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),P2: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,P2,bot_bot(set(A)))
           => ( ! [B4: A,A6: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A6)
                 => ( ! [X: A] :
                        ( member(A,X,A6)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B4) )
                   => ( aa(set(A),$o,P2,A6)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),A6)) ) ) )
             => aa(set(A),$o,P2,A5) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2333_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),P2: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,P2,bot_bot(set(A)))
           => ( ! [B4: A,A6: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A6)
                 => ( ! [X: A] :
                        ( member(A,X,A6)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),X) )
                   => ( aa(set(A),$o,P2,A6)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),A6)) ) ) )
             => aa(set(A),$o,P2,A5) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2334_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = $ite(member(A,Xb,A5),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5))) ) ) ) ).

% sum.insert_if
tff(fact_2335_finite__subset__induct,axiom,
    ! [A: $tType,F4: set(A),A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A5)
       => ( aa(set(A),$o,P2,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( member(A,A4,A5)
                 => ( ~ member(A,A4,F5)
                   => ( aa(set(A),$o,P2,F5)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),F5)) ) ) ) )
           => aa(set(A),$o,P2,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_2336_finite__subset__induct_H,axiom,
    ! [A: $tType,F4: set(A),A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A5)
       => ( aa(set(A),$o,P2,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( member(A,A4,A5)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A5)
                   => ( ~ member(A,A4,F5)
                     => ( aa(set(A),$o,P2,F5)
                       => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),F5)) ) ) ) ) )
           => aa(set(A),$o,P2,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2337_finite__empty__induct,axiom,
    ! [A: $tType,A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,P2,A5)
       => ( ! [A4: A,A6: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A6)
             => ( member(A,A4,A6)
               => ( aa(set(A),$o,P2,A6)
                 => aa(set(A),$o,P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P2,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_2338_infinite__coinduct,axiom,
    ! [A: $tType,X5: fun(set(A),$o),A5: set(A)] :
      ( aa(set(A),$o,X5,A5)
     => ( ! [A6: set(A)] :
            ( aa(set(A),$o,X5,A6)
           => ? [X: A] :
                ( member(A,X,A6)
                & ( aa(set(A),$o,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))
                  | ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% infinite_coinduct
tff(fact_2339_infinite__remove,axiom,
    ! [A: $tType,S2: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ~ aa(set(A),$o,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)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_2340_subset__insert__iff,axiom,
    ! [A: $tType,A5: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),B5))
    <=> $ite(member(A,Xb,A5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)) ) ).

% subset_insert_iff
tff(fact_2341_Diff__single__insert,axiom,
    ! [A: $tType,A5: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),B5)) ) ).

% Diff_single_insert
tff(fact_2342_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_2343_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Mb),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) = set_or1337092689740270186AtMost(nat,Mb,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_2344_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
     => ( set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2345_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( set_or1337092689740270186AtMost(nat,Mb,Nb) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Mb),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2346_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,Xb))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_2347_Compl__insert,axiom,
    ! [A: $tType,Xb: A,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = 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)),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_2348_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,I6: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ck(A,fun(nat,fun(nat,A)),Xb),Mb)),I6) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Xb),Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),I6)) ) ).

% sum_power_add
tff(fact_2349_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_cl(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2350_less__mask,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),bit_se2239418461657761734s_mask(nat,Nb)) ) ).

% less_mask
tff(fact_2351_sum__nth__roots,axiom,
    ! [Nb: nat,C3: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_cm(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_aw(nat,fun(complex,fun(complex,$o)),Nb),C3))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_2352_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_cm(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_cn(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_2353_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_2354_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,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_2355_sum__diff__nat,axiom,
    ! [A: $tType,B5: set(A),A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),B5)) ) ) ) ).

% sum_diff_nat
tff(fact_2356_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,zero_zero(nat),K2)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_2357_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2358_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2359_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2360_remove__induct,axiom,
    ! [A: $tType,P2: fun(set(A),$o),B5: set(A)] :
      ( aa(set(A),$o,P2,bot_bot(set(A)))
     => ( ( ~ aa(set(A),$o,finite_finite2(A),B5)
         => aa(set(A),$o,P2,B5) )
       => ( ! [A6: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A6)
             => ( ( A6 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),B5)
                 => ( ! [X: A] :
                        ( member(A,X,A6)
                       => aa(set(A),$o,P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) )
                   => aa(set(A),$o,P2,A6) ) ) ) )
         => aa(set(A),$o,P2,B5) ) ) ) ).

% remove_induct
tff(fact_2361_finite__remove__induct,axiom,
    ! [A: $tType,B5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,P2,bot_bot(set(A)))
       => ( ! [A6: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A6)
             => ( ( A6 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),B5)
                 => ( ! [X: A] :
                        ( member(A,X,A6)
                       => aa(set(A),$o,P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) )
                   => aa(set(A),$o,P2,A6) ) ) ) )
         => aa(set(A),$o,P2,B5) ) ) ) ).

% finite_remove_induct
tff(fact_2362_finite__induct__select,axiom,
    ! [A: $tType,S2: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(A),$o,P2,bot_bot(set(A)))
       => ( ! [T4: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T4),S2)
             => ( aa(set(A),$o,P2,T4)
               => ? [X: A] :
                    ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T4))
                    & aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),T4)) ) ) )
         => aa(set(A),$o,P2,S2) ) ) ) ).

% finite_induct_select
tff(fact_2363_psubset__insert__iff,axiom,
    ! [A: $tType,A5: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),B5))
    <=> $ite(
          member(A,Xb,B5),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5),
          $ite(member(A,Xb,A5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)) ) ) ).

% psubset_insert_iff
tff(fact_2364_set__replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Nb),Xb)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_2365_set__replicate__conv__if,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      aa(list(A),set(A),set2(A),replicate(A,Nb,Xb)) = $ite(Nb = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_2366_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ci(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2367_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,aa(nat,nat,suc,Nb))),aa(nat,A,F3,Mb)) ) ) ) ).

% sum_Suc_diff
tff(fact_2368_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P2: fun(int,$o),Ta: A] :
          ( aa(int,$o,P2,archimedean_ceiling(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),aa(int,A,ring_1_of_int(A),I4)) )
             => aa(int,$o,P2,I4) ) ) ) ).

% ceiling_split
tff(fact_2369_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archimedean_ceiling(A,Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_2370_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xb) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_2371_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))),one_one(A))),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ) ).

% ceiling_correct
tff(fact_2372_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),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_2373_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_2374_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb) ) ) ).

% le_ceiling_iff
tff(fact_2375_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A),P: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P))) = 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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2376_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),Xb: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),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)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_2377_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),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)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_2378_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),F3: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),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)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),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),F3),A5)),aa(A,B,F3,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)) ) ) ) ).

% sum_diff1
tff(fact_2379_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X5) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_2380_set__encode__def,axiom,
    nat_set_encode = aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% set_encode_def
tff(fact_2381_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_cp(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C3)),S2) = $ite(member(A,A2,S2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C3),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)),insert(A),A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C3),aa(set(A),set(A),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)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_2382_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P),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,P,Q2)))),Q2)) ) ) ).

% ceiling_divide_upper
tff(fact_2383_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2239418461657761734s_mask(nat,Nb)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% mask_nat_less_exp
tff(fact_2384_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I2: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I2,A5)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3)) )
           => ( aa(set(A),$o,finite_finite2(A),A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)) ) ) ) ) ).

% member_le_sum
tff(fact_2385_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cq(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Mb)),aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_2386_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cr(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,Mb)) ) ) ) ).

% sum_telescope''
tff(fact_2387_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,Nb))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_2388_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ member(nat,Nb,nat_set_decode(Z))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Z)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Nb),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_2389_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P,Q2)))),one_one(A))),Q2)),P) ) ) ).

% ceiling_divide_lower
tff(fact_2390_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Nb)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Nb)),one_one(A)))
           => ( archimedean_ceiling(A,Xb) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_2391_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb))) ) ).

% mask_eq_sum_exp
tff(fact_2392_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Mb: nat,Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),set_or1337092689740270186AtMost(nat,Mb,Nb))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,Xb),Mb)),aa(nat,A,power_power(A,Xb),aa(nat,nat,suc,Nb))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2393_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cs(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.in_pairs
tff(fact_2394_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_2395_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ca(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_2396_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),
            zero_zero(A),
            $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))),Mb)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,Xb),Mb)),aa(nat,A,power_power(A,Xb),aa(nat,nat,suc,Nb))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ) ).

% sum_gp
tff(fact_2397_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: 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)),Nb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2398_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) = $ite(Xb = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Xb),Mb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,Xb),aa(nat,nat,suc,Nb)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp_offset
tff(fact_2399_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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)),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_2400_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: 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),Nb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_2401_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D3: A,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ct(A,fun(A,fun(nat,A)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D3))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_2402_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Mb: nat,Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Mb) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( Mb = Nb ) ) ) ).

% of_nat_eq_iff
tff(fact_2403_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Mb) = aa(num,int,numeral_numeral(int),V) )
    <=> ( Mb = aa(num,nat,numeral_numeral(nat),V) ) ) ).

% int_eq_iff_numeral
tff(fact_2404_negative__eq__positive,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,int,semiring_1_of_nat(int),Mb) )
    <=> ( ( Nb = zero_zero(nat) )
        & ( Mb = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_2405_of__int__of__nat__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ).

% of_int_of_nat_eq
tff(fact_2406_negative__zle,axiom,
    ! [Nb: nat,Mb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb))),aa(nat,int,semiring_1_of_nat(int),Mb)) ).

% negative_zle
tff(fact_2407_int__dvd__int__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ).

% int_dvd_int_iff
tff(fact_2408_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Mb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Mb) = zero_zero(A) )
        <=> ( Mb = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_2409_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( zero_zero(nat) = Nb ) ) ) ).

% of_nat_0_eq_iff
tff(fact_2410_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_2411_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% of_nat_less_iff
tff(fact_2412_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% of_nat_le_iff
tff(fact_2413_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_add
tff(fact_2414_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_mult
tff(fact_2415_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_2416_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( Nb = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_2417_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Nb) = one_one(A) )
        <=> ( Nb = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_2418_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)))),aa(nat,int,semiring_1_of_nat(int),Mb)) ).

% negative_zless
tff(fact_2419_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P2: $o] : aa(nat,A,semiring_1_of_nat(A),aa($o,nat,zero_neq_one_of_bool(nat),(P2))) = aa($o,A,zero_neq_one_of_bool(A),(P2)) ) ).

% of_nat_of_bool
tff(fact_2420_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F3: fun(B,nat),A5: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cu(fun(B,nat),fun(B,A),F3)),A5) ) ).

% of_nat_sum
tff(fact_2421_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Mb)),zero_zero(A))
        <=> ( Mb = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_2422_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Mb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Mb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Mb)) ) ).

% of_nat_Suc
tff(fact_2423_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(num,real,numeral_numeral(real),W2)),aa(nat,real,semiring_1_of_nat(real),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),W2)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_2424_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W2: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(num,real,numeral_numeral(real),W2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_2425_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Mb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_2426_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% of_nat_0_less_iff
tff(fact_2427_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B2: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,B2),W2)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_2428_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),W2)),Xb) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_2429_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B2: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(nat,nat,power_power(nat,B2),W2)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2430_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),W2)),Xb) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2431_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),Xb)),Nb))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xb)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2432_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I2: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),Nb)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2433_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),Nb)),Xb) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2434_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I2: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),Nb)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2435_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),Nb)),Xb) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2436_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% real_arch_simple
tff(fact_2437_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% reals_Archimedean2
tff(fact_2438_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: nat,Yb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xb)),Yb) = aa(A,A,aa(A,fun(A,A),times_times(A),Yb),aa(nat,A,semiring_1_of_nat(A),Xb)) ) ).

% mult_of_nat_commute
tff(fact_2439_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% int_cases2
tff(fact_2440_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N2: nat] : Z != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_diff_cases
tff(fact_2441_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Nb)),Xb) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2442_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_0_le_iff
tff(fact_2443_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Mb)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_2444_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Nb)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_2445_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% of_nat_less_imp_less
tff(fact_2446_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% less_imp_of_nat_less
tff(fact_2447_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I2: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I2)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_2448_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_2449_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),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_2450_int__of__nat__induct,axiom,
    ! [P2: fun(int,$o),Z: int] :
      ( ! [N2: nat] : aa(int,$o,P2,aa(nat,int,semiring_1_of_nat(int),N2))
     => ( ! [N2: nat] : aa(int,$o,P2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))))
       => aa(int,$o,P2,Z) ) ) ).

% int_of_nat_induct
tff(fact_2451_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% int_cases
tff(fact_2452_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),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_2453_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% zle_int
tff(fact_2454_zero__le__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ? [N2: nat] : K2 = aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% zero_le_imp_eq_int
tff(fact_2455_nonneg__int__cases,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ~ ! [N2: nat] : K2 != aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% nonneg_int_cases
tff(fact_2456_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_2457_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(nat,int,semiring_1_of_nat(int),Mb)) ).

% int_plus
tff(fact_2458_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_2459_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z)
    <=> ? [N: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% zle_iff_zadd
tff(fact_2460_not__int__zless__negative,axiom,
    ! [Nb: nat,Mb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Mb))) ).

% not_int_zless_negative
tff(fact_2461_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Yb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,semiring_1_of_nat(A),Yb)) ) ).

% of_nat_max
tff(fact_2462_nat__less__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_2463_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_2464_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),Xb)) ) ) ).

% ex_less_of_nat_mult
tff(fact_2465_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Nb: nat,Mb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ) ).

% of_nat_diff
tff(fact_2466_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ! [Y3: real] :
        ? [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),Xb)) ) ).

% reals_Archimedean3
tff(fact_2467_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N2: nat] : Mb != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( Mb != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_2468_int__zle__neg,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Mb)))
    <=> ( ( Nb = zero_zero(nat) )
        & ( Mb = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2469_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
    <=> ? [N: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% zless_iff_Suc_zadd
tff(fact_2470_negative__zle__0,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb))),zero_zero(int)) ).

% negative_zle_0
tff(fact_2471_nonpos__int__cases,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),zero_zero(int))
     => ~ ! [N2: nat] : K2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% nonpos_int_cases
tff(fact_2472_int__sum,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7311177749621191930dd_sum(A,int),aTP_Lamp_cv(fun(A,nat),fun(A,int),F3)),A5) ).

% int_sum
tff(fact_2473_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,Mb: nat,Nb: nat] : modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),modulo_modulo(A,divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),Mb))) ) ).

% mod_mult2_eq'
tff(fact_2474_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
          & ( K2 = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2475_pos__int__cases,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ~ ! [N2: nat] :
            ( ( K2 = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ).

% pos_int_cases
tff(fact_2476_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K2 = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) )
       => ~ ! [N2: nat] :
              ( ( K2 = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ) ).

% int_cases3
tff(fact_2477_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Nb)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),Mb)) ) ).

% nat_less_real_le
tff(fact_2478_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Mb)),one_one(real))) ) ).

% nat_le_real_less
tff(fact_2479_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K2: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K2)),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K2)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_2480_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)))) ).

% not_zle_0_negative
tff(fact_2481_negative__zless__0,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_2482_negD,axiom,
    ! [Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),zero_zero(int))
     => ? [N2: nat] : Xb = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_2483_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ).

% int_ops(6)
tff(fact_2484_atLeastAtMostPlus1__int__conv,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Mb),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb))
     => ( set_or1337092689740270186AtMost(int,Mb,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)),set_or1337092689740270186AtMost(int,Mb,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2485_simp__from__to,axiom,
    ! [I2: int,J: int] :
      set_or1337092689740270186AtMost(int,I2,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I2),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),I2),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J))) ).

% simp_from_to
tff(fact_2486_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E3)
         => ~ ! [N2: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),E3) ) ) ).

% nat_approx_posE
tff(fact_2487_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% of_nat_less_two_power
tff(fact_2488_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: nat,Mb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => ( ( Nb != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Mb))),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ) ) ).

% inverse_of_nat_le
tff(fact_2489_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C3)
       => ( ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),Xb)),C3) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2490_neg__int__cases,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
     => ~ ! [N2: nat] :
            ( ( K2 = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ).

% neg_int_cases
tff(fact_2491_zdiff__int__split,axiom,
    ! [P2: fun(int,$o),Xb: nat,Yb: nat] :
      ( aa(int,$o,P2,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Yb)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Yb),Xb)
         => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Xb)),aa(nat,int,semiring_1_of_nat(int),Yb))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Yb)
         => aa(int,$o,P2,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_2492_ln__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(nat,real,power_power(real,Xb),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_realpow
tff(fact_2493_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_2494_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D3: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_cw(A,fun(A,fun(nat,A)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D3))) ) ).

% double_arith_series
tff(fact_2495_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(nat,A,semiring_1_of_nat(A),Nb) = $ite(Nb = zero_zero(nat),zero_zero(A),aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_cx(nat,fun(nat,A))),divmod_nat(Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_2496_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K5: real,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K5)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Nb)),aa(nat,A,power_power(A,Z),Nb)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,power_power(A,Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,power_power(real,K5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2497_ln__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),Xb) = suminf(real,aTP_Lamp_cy(real,fun(nat,real),Xb)) ) ) ) ).

% ln_series
tff(fact_2498_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Nb)),aa(nat,A,power_power(A,Z),Nb)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,power_power(A,Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,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_da(A,fun(A,fun(nat,fun(nat,A))),H),Z),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_2499_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),comm_s3205402744901411588hammer(A,Z,Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)) ) ).

% pochhammer_double
tff(fact_2500_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),Nb) = semiri8178284476397505188at_aux(A,aTP_Lamp_db(A,A),Nb,zero_zero(A)) ) ).

% of_nat_code
tff(fact_2501_lessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),Xb) = aa(A,set(A),set_ord_lessThan(A),Yb) )
        <=> ( Xb = Yb ) ) ) ).

% lessThan_eq_iff
tff(fact_2502_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( member(A,I2,aa(A,set(A),set_ord_lessThan(A),K2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),K2) ) ) ).

% lessThan_iff
tff(fact_2503_finite__lessThan,axiom,
    ! [K2: nat] : aa(set(nat),$o,finite_finite2(nat),aa(nat,set(nat),set_ord_lessThan(nat),K2)) ).

% finite_lessThan
tff(fact_2504_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),Xb)),aa(A,set(A),set_ord_lessThan(A),Yb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ).

% lessThan_subset_iff
tff(fact_2505_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_2506_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_2507_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_2508_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: 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)),insert(A),K2),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K2),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_2509_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = 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),G3),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(nat,A,G3,Nb)) ) ).

% sum.lessThan_Suc
tff(fact_2510_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F3: fun(nat,A)] : suminf(A,aTP_Lamp_dc(fun(nat,A),fun(nat,A),F3)) = aa(nat,A,F3,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2511_int__int__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Mb) = aa(nat,int,semiring_1_of_nat(int),Nb) )
    <=> ( Mb = Nb ) ) ).

% int_int_eq
tff(fact_2512_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] : aa(A,set(A),set_ord_lessThan(A),Xb) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_2513_infinite__Iio,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A2: A] : ~ aa(set(A),$o,finite_finite2(A),aa(A,set(A),set_ord_lessThan(A),A2)) ) ).

% infinite_Iio
tff(fact_2514_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_dd(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_2515_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [Nb: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),Nb) = bot_bot(set(A)) )
        <=> ( Nb = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_2516_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Mb: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),Mb)),aa(A,set(A),set_ord_lessThan(A),Nb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb) ) ) ).

% lessThan_strict_subset_iff
tff(fact_2517_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,Nb)) ) ) ).

% pochhammer_pos
tff(fact_2518_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat,Mb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( comm_s3205402744901411588hammer(A,A2,Mb) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2519_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Mb) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( comm_s3205402744901411588hammer(A,A2,Nb) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_2520_lessThan__Suc,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),K2),aa(nat,set(nat),set_ord_lessThan(nat),K2)) ).

% lessThan_Suc
tff(fact_2521_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),Nb) = bot_bot(set(nat)) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_2522_finite__nat__bounded,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => ? [K: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_lessThan(nat),K)) ) ).

% finite_nat_bounded
tff(fact_2523_finite__nat__iff__bounded,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_lessThan(nat),K3)) ) ).

% finite_nat_iff_bounded
tff(fact_2524_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,Nb)) ) ) ).

% pochhammer_nonneg
tff(fact_2525_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Nb: nat,I2: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Nb),I2) = semiri8178284476397505188at_aux(A,Inc,Nb,aa(A,A,Inc,I2)) ) ).

% of_nat_aux.simps(2)
tff(fact_2526_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I2: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I2) = I2 ) ).

% of_nat_aux.simps(1)
tff(fact_2527_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] :
          comm_s3205402744901411588hammer(A,zero_zero(A),Nb) = $ite(Nb = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% pochhammer_0_left
tff(fact_2528_lessThan__nat__numeral,axiom,
    ! [K2: num] : aa(nat,set(nat),set_ord_lessThan(nat),aa(num,nat,numeral_numeral(nat),K2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),pred_numeral(K2)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K2))) ).

% lessThan_nat_numeral
tff(fact_2529_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,fun(nat,A)),G3),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.nat_diff_reindex
tff(fact_2530_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P2: fun(A,nat),Nb: A] :
          ( ! [X3: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X3)),aa(A,nat,P2,X3))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P2),aa(A,set(A),set_ord_lessThan(A),Nb))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),aa(A,set(A),set_ord_lessThan(A),Nb))) = 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_df(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P2)),aa(A,set(A),set_ord_lessThan(A),Nb)) ) ) ) ).

% sum_diff_distrib
tff(fact_2531_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N))),K6) )
        <=> ? [N6: nat] :
            ! [N: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% lemma_NBseq_def
tff(fact_2532_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N))),K6) )
        <=> ? [N6: nat] :
            ! [N: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% lemma_NBseq_def2
tff(fact_2533_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),Nb)) ) ).

% pochhammer_rec
tff(fact_2534_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ).

% pochhammer_Suc
tff(fact_2535_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Nb: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Nb))),comm_s3205402744901411588hammer(A,Z,Nb)) ) ).

% pochhammer_rec'
tff(fact_2536_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K2) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2537_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Nb: nat,K2: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K2) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2538_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Nb)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2539_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K2) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2540_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Nb: nat,Mb: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,Nb)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Nb)),Mb)) ) ).

% pochhammer_product'
tff(fact_2541_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ci(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2542_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Mb)),aa(nat,A,F3,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2543_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dg(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,Mb)) ) ).

% sum_lessThan_telescope'
tff(fact_2544_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ci(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2545_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,Xb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% one_diff_power_eq
tff(fact_2546_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,Xb),Nb)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% power_diff_1_eq
tff(fact_2547_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Nb: nat] :
          ( ( Xb != one_one(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,Xb),Nb)),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_2548_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Mb: nat,Nb: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( comm_s3205402744901411588hammer(A,Z,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,Mb)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ) ) ).

% pochhammer_product
tff(fact_2549_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,Xb),Nb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp_strict
tff(fact_2550_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dh(A,fun(A,fun(nat,fun(nat,A))),Z),H),Mb)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) = 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_di(A,fun(A,fun(nat,fun(nat,A))),Z),H),Mb)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) ) ).

% lemma_termdiff1
tff(fact_2551_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat,Yb: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,Xb),Nb)),aa(nat,A,power_power(A,Yb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dj(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Yb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% power_diff_sumr2
tff(fact_2552_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat,Yb: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,Xb),aa(nat,nat,suc,Nb))),aa(nat,A,power_power(A,Yb),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dk(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Yb)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb)))) ) ).

% diff_power_eq_sum
tff(fact_2553_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(nat,A,semiring_1_of_nat(A),K2))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K2)) ) ).

% pochhammer_absorb_comp
tff(fact_2554_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F3: fun(nat,A),K5: A,K2: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,P4)),K5) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K5)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_2555_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,Xb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dl(A,fun(nat,fun(nat,A)),Xb),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% one_diff_power_eq'
tff(fact_2556_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A)),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K2)) ) ).

% pochhammer_minus'
tff(fact_2557_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A)),K2)) ) ).

% pochhammer_minus
tff(fact_2558_sum__split__even__odd,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real),Nb: nat] : aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_dm(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F3),G3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) = 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_dn(fun(nat,real),fun(nat,real),F3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_do(fun(nat,real),fun(nat,real),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ).

% sum_split_even_odd
tff(fact_2559_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xb)),zero_zero(real))
        <=> ( Xb = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2560_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xb))
        <=> ( Xb != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2561_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => ( suminf(A,power_power(A,C3)) = divide_divide(A,one_one(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C3)) ) ) ) ).

% suminf_geometric
tff(fact_2562_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2563_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( ( real_V7770717601297561774m_norm(A,Xb) = zero_zero(real) )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2564_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_dp(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_2565_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_2566_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Yb: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xb)),Yb)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) ) ).

% norm_uminus_minus
tff(fact_2567_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_2568_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,Nb: nat,Z: A] :
          ( ( aa(nat,A,power_power(A,W2),Nb) = aa(nat,A,power_power(A,Z),Nb) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( real_V7770717601297561774m_norm(A,W2) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2569_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,R2: real,Yb: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Yb)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S)) ) ) ) ).

% norm_mult_less
tff(fact_2570_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Yb: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Yb))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb))),E3) ) ) ).

% norm_triangle_lt
tff(fact_2571_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,R2: real,Yb: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Yb)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S)) ) ) ) ).

% norm_add_less
tff(fact_2572_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),C3)) ) ) ).

% norm_add_leD
tff(fact_2573_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Yb: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Yb))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb))),E3) ) ) ).

% norm_triangle_le
tff(fact_2574_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Yb: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Yb))) ) ).

% norm_triangle_ineq
tff(fact_2575_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R2: real,B2: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S)) ) ) ) ).

% norm_triangle_mono
tff(fact_2576_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Yb: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Yb),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2577_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ).

% norm_diff_ineq
tff(fact_2578_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N3: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N3)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,N3)
               => ( aa(nat,A,F3,N2) = zero_zero(A) ) )
           => ( suminf(A,F3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),N3) ) ) ) ) ).

% suminf_finite
tff(fact_2579_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,Nb: nat] :
          ( ( aa(nat,A,power_power(A,W2),Nb) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W2) = one_one(real) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2580_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C3: A,D3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D3)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2581_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dq(A,fun(nat,A),Z)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_2582_pi__series,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_dr(nat,real)) ).

% pi_series
tff(fact_2583_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = $ite(
        ( member(int,K2,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)
            & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
        aa(int,int,
          aa(int,fun(int,int),plus_plus(int),
            aa($o,int,zero_neq_one_of_bool(int),
              ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)
              & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% and_int.simps
tff(fact_2584_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Yb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Yb )
     => ( Yb = $ite(
            ( member(int,Xb,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                  & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,Xb,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xaa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.elims
tff(fact_2585_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] :
          comm_s3205402744901411588hammer(A,A2,Nb) = $ite(Nb = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_ds(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_2586_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),Nb)),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_2587_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_2588_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_2589_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xb) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_2590_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_2591_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,nat),A5: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_dt(fun(B,nat),fun(B,A),F3)),A5) ) ).

% of_nat_prod
tff(fact_2592_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F3: fun(B,int),A5: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F3),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_du(fun(B,int),fun(B,A),F3)),A5) ) ).

% of_int_prod
tff(fact_2593_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5) = zero_zero(B) )
          <=> ? [X4: A] :
                ( member(A,X4,A5)
                & ( aa(A,B,F3,X4) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_2594_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_2595_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_2596_dvd__prod__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A5: set(A),A2: A,B2: B,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ( ( B2 = aa(A,B,F3,A2) )
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_2597_dvd__prodI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A5: set(A),A2: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F3,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ) ).

% dvd_prodI
tff(fact_2598_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_2599_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_dv(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_2600_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_dw(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_2601_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),Xb: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ) ) ).

% prod.insert
tff(fact_2602_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(nat,A,G3,Nb)) ) ).

% prod.lessThan_Suc
tff(fact_2603_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,log(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_2604_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xb)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_2605_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,log(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_2606_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_2607_log__less__cancel__iff,axiom,
    ! [A2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Yb))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_2608_log__eq__one,axiom,
    ! [A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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_2609_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xb))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_2610_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Yb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Yb))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_2611_log__le__cancel__iff,axiom,
    ! [A2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Yb))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Yb) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_2612_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_2613_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,log(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_2614_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xb)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_2615_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,log(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_2616_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb)))) ) ).

% prod.cl_ivl_Suc
tff(fact_2617_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),aa(nat,real,power_power(real,A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_2618_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Yb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit1(Xb))),aa(num,A,numeral_numeral(A),bit1(Yb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Yb)))) ) ).

% and_numerals(7)
tff(fact_2619_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A5: set(A),B5: set(B),G3: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(B),$o,finite_finite2(B),B5)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_dx(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B5),G3),R)),A5) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_dz(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A5),G3),R)),B5) ) ) ) ) ).

% prod.swap_restrict
tff(fact_2620_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I3))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),aa(A,B,G3,I3)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ).

% prod_mono
tff(fact_2621_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ).

% prod_nonneg
tff(fact_2622_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ).

% prod_pos
tff(fact_2623_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ).

% prod_ge_1
tff(fact_2624_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ? [X: A] :
                ( member(A,X,A5)
                & ( aa(A,B,F3,X) = zero_zero(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_2625_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F3: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F3),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ea(fun(nat,A),fun(nat,fun(A,A)),F3),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_2626_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_2627_pi__not__less__zero,axiom,
    ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_2628_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(set(A),fun(fun(A,$o),fun(A,$o)),A5),P2))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_eb(fun(A,B),fun(fun(A,$o),fun(A,B)),G3),P2)),A5) ) ) ) ).

% prod.inter_filter
tff(fact_2629_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ec(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_2630_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ed(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_2631_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_2632_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S2: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X15: A,Y15: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),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),X22),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S2)
             => ( ! [X3: B] :
                    ( member(B,X3,S2)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X3)),aa(B,A,G3,X3)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S2)) ) ) ) ) ) ).

% prod.related
tff(fact_2633_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = $ite(member(A,Xb,A5),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5))) ) ) ) ).

% prod.insert_if
tff(fact_2634_prod__dvd__prod__subset2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F3,A4)),aa(A,B,G3,A4)) )
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_2635_prod__dvd__prod__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B5)) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_2636_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T3: set(B),S2: set(A),I2: fun(B,A),J: fun(A,B),T2: set(B),G3: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5))
                 => ( aa(B,A,I2,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                     => ( aa(A,B,J,aa(B,A,I2,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                       => member(A,aa(B,A,I2,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S5)
                         => ( aa(A,C,G3,A4) = one_one(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T3)
                           => ( aa(B,C,H,B4) = one_one(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S2)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G3,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G3),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),T2) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_2637_AND__upper2_H_H,axiom,
    ! [Yb: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Yb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Yb),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Yb)),Z) ) ) ).

% AND_upper2''
tff(fact_2638_AND__upper1_H_H,axiom,
    ! [Yb: int,Z: int,Yaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Yb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Yb),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Yb),Yaa)),Z) ) ) ).

% AND_upper1''
tff(fact_2639_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L)),K2) ) ).

% and_less_eq
tff(fact_2640_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ee(fun(A,B),fun(A,$o),G3)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_2641_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ef(fun(nat,A),fun(nat,fun(nat,A)),G3),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.nat_diff_reindex
tff(fact_2642_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_eg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% prod.atLeastAtMost_rev
tff(fact_2643_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),I2: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( member(A,I2,I6)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F3,I2))
             => ( ! [I3: A] :
                    ( member(A,I3,I6)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I6)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_2644_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ( I6 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I6)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F3,I3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I6)) ) ) ) ) ).

% less_1_prod
tff(fact_2645_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(A),A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ).

% prod.subset_diff
tff(fact_2646_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5))
                   => ( aa(A,B,G3,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B5))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B5) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_2647_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5))
                   => ( aa(A,B,G3,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B5))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C5) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B5) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_2648_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,G3,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T2) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_2649_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,G3,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S2) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_2650_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S2: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,H,X3) = one_one(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_2651_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S2: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,G3,X3) = one_one(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_2652_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_2653_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_2654_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_2655_log__base__change,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),Xb) = divide_divide(real,aa(real,real,log(A2),Xb),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_2656_log__of__power__eq,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),Mb) = aa(nat,real,power_power(real,B2),Nb) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,real,semiring_1_of_nat(real),Nb) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_2657_less__log__of__power,axiom,
    ! [B2: real,Nb: nat,Mb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,B2),Nb)),Mb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Mb)) ) ) ).

% less_log_of_power
tff(fact_2658_pi__less__4,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% pi_less_4
tff(fact_2659_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ec(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% prod.lessThan_Suc_shift
tff(fact_2660_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ec(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_2661_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ec(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_2662_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [I3: A] :
                ( member(A,I3,A5)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I3))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,I3)),aa(A,B,G3,I3)) ) )
           => ( ( A5 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ) ) ).

% prod_mono_strict
tff(fact_2663_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5))
          <=> ? [X4: A] :
                ( member(A,X4,A5)
                & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,F3,X4)) ) ) ) ) ).

% even_prod_iff
tff(fact_2664_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),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)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_2665_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),Xb: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),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)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_2666_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A),P: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P))) = 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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_2667_log__mult,axiom,
    ! [A2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Yb)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Yb)) ) ) ) ) ) ).

% log_mult
tff(fact_2668_log__divide,axiom,
    ! [A2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
           => ( aa(real,real,log(A2),divide_divide(real,Xb,Yb)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Yb)) ) ) ) ) ) ).

% log_divide
tff(fact_2669_le__log__of__power,axiom,
    ! [B2: real,Nb: nat,Mb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,B2),Nb)),Mb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Mb)) ) ) ).

% le_log_of_power
tff(fact_2670_log__base__pow,axiom,
    ! [A2: real,Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,real,log(aa(nat,real,power_power(real,A2),Nb)),Xb) = divide_divide(real,aa(real,real,log(A2),Xb),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log_base_pow
tff(fact_2671_log__nat__power,axiom,
    ! [Xb: real,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log(B2),aa(nat,real,power_power(real,Xb),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Xb)) ) ) ).

% log_nat_power
tff(fact_2672_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_eh(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C3)),S2) = $ite(member(A,A2,S2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C3),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)),insert(A),A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C3),aa(set(A),set(A),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)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_2673_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xb: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xc: A,Yb: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xb,Xaa,Xba,Xc) = Yb )
     => ( Yb = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xba),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xba,aa(A,A,aa(nat,fun(A,A),Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2674_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A2),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2675_pi__half__less__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_less_two
tff(fact_2676_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,B4)) )
             => ( ! [A4: A] :
                    ( member(A,A4,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,A4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B5)) ) ) ) ) ) ).

% prod_mono2
tff(fact_2677_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A5: set(A),F3: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( aa(A,B,F3,A2) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),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)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5),aa(A,B,F3,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ) ) ).

% prod_diff1
tff(fact_2678_log__of__power__less,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Mb)),aa(nat,real,power_power(real,B2),Nb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),Mb))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_less
tff(fact_2679_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
         => ( ( B2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
             => ( aa(real,real,log(A2),Xb) = 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),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2680_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ei(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod
tff(fact_2681_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_gt_zero
tff(fact_2682_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ej(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ).

% pochhammer_prod_rev
tff(fact_2683_m2pi__less__pi,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))),pi) ).

% m2pi_less_pi
tff(fact_2684_log__of__power__le,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Mb)),aa(nat,real,power_power(real,B2),Nb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),Mb))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_le
tff(fact_2685_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ek(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.in_pairs
tff(fact_2686_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_el(fun(nat,A),fun(nat,fun(A,A)),F3),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_2687_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ej(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_2688_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_2689_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Mb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Mb))) ) ).

% less_log2_of_power
tff(fact_2690_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Mb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Mb))) ) ).

% le_log2_of_power
tff(fact_2691_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Mb))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_less
tff(fact_2692_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Mb))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_le
tff(fact_2693_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),Nb)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),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),K2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2694_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),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),Nb),one_one(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_2695_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K2: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),Xb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K2)),one_one(int)) )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K2))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_2696_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => sums(A,aTP_Lamp_em(A,fun(nat,A),Z),divide_divide(A,one_one(A),aa(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),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_2697_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K2))) = aa(nat,int,semiring_1_of_nat(int),Nb) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),Nb)),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_2698_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Yb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Yb )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xb),Xaa))
       => ~ ( ( Yb = $ite(
                  ( member(int,Xb,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                      & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
                  aa(int,int,
                    aa(int,fun(int,int),plus_plus(int),
                      aa($o,int,zero_neq_one_of_bool(int),
                        ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                        & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,Xb,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,Xaa,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_2699_and__int_Opsimps,axiom,
    ! [K2: 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),K2),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = $ite(
            ( member(int,K2,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)
                & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K2)
                  & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K2,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_2700_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Nb)))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_2701_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_2702_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_2703_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_en(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_2704_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] :
          powr(A,Xb,zero_zero(A)) = $ite(Xb = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_2705_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_2706_powr__gt__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),powr(real,Xb,A2))
    <=> ( Xb != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_2707_powr__less__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_2708_prod__eq__1__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A5) = one_one(nat) )
      <=> ! [X4: A] :
            ( member(A,X4,A5)
           => ( aa(A,nat,F3,X4) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_2709_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_2710_and__nat__numerals_I1_J,axiom,
    ! [Yb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Yb))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_2711_powr__eq__one__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( ( powr(real,A2,Xb) = one_one(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_2712_powr__le__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_2713_prod__pos__nat__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A5))
      <=> ! [X4: A] :
            ( member(A,X4,A5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_2714_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit1(Xb))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_2715_and__nat__numerals_I2_J,axiom,
    ! [Yb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Yb))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_2716_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) ) ).

% zero_le_floor
tff(fact_2717_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_2718_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V)),Xb) ) ) ).

% numeral_le_floor
tff(fact_2719_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ).

% zero_less_floor
tff(fact_2720_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_2721_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(num,A,numeral_numeral(A),V)) ) ) ).

% floor_less_numeral
tff(fact_2722_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ).

% one_le_floor
tff(fact_2723_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ).

% floor_less_one
tff(fact_2724_log__powr__cancel,axiom,
    ! [A2: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),powr(real,A2,Yb)) = Yb ) ) ) ).

% log_powr_cancel
tff(fact_2725_powr__log__cancel,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( powr(real,A2,aa(real,real,log(A2),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_2726_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xb: A] :
          ( sums(A,aTP_Lamp_eo(fun(nat,A),fun(nat,A),A2),Xb)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xb ) ) ) ).

% powser_sums_zero_iff
tff(fact_2727_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_2728_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_2729_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),Xb) ) ) ).

% numeral_less_floor
tff(fact_2730_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_2731_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb) ) ) ).

% one_less_floor
tff(fact_2732_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_2733_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xb) ) ) ).

% neg_numeral_le_floor
tff(fact_2734_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_neg_numeral
tff(fact_2735_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),Xb) ) ) ).

% neg_numeral_less_floor
tff(fact_2736_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_2737_int__prod,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A5)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7121269368397514597t_prod(A,int),aTP_Lamp_cv(fun(A,nat),fun(A,int),F3)),A5) ).

% int_prod
tff(fact_2738_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A)] :
          ( ! [N2: nat] : aa(nat,A,F3,N2) = zero_zero(A)
         => sums(A,F3,zero_zero(A)) ) ) ).

% sums_0
tff(fact_2739_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A),S: A,Ta: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,G3,N2))
         => ( sums(A,F3,S)
           => ( sums(A,G3,Ta)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S),Ta) ) ) ) ) ).

% sums_le
tff(fact_2740_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I2: nat,F3: fun(nat,A)] : sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ep(nat,fun(fun(nat,A),fun(nat,A)),I2),F3),aa(nat,A,F3,I2)) ) ).

% sums_single
tff(fact_2741_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),A2: A,G3: fun(nat,A),B2: A] :
          ( sums(A,F3,A2)
         => ( sums(A,G3,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% sums_add
tff(fact_2742_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Yb)) ) ) ).

% floor_mono
tff(fact_2743_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb))),Xb) ) ).

% of_int_floor_le
tff(fact_2744_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Yb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb) ) ) ).

% floor_less_cancel
tff(fact_2745_powr__less__mono2__neg,axiom,
    ! [A2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Yb,A2)),powr(real,Xb,A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_2746_powr__non__neg,axiom,
    ! [A2: real,Xb: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,A2,Xb)),zero_zero(real)) ).

% powr_non_neg
tff(fact_2747_powr__less__cancel,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel
tff(fact_2748_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2)) ) ) ).

% powr_less_mono
tff(fact_2749_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F3: fun(nat,A),D3: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_er(A,fun(fun(nat,A),fun(nat,A)),C3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D3))
          <=> sums(A,F3,D3) ) ) ) ).

% sums_mult_iff
tff(fact_2750_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F3: fun(nat,A),D3: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_es(A,fun(fun(nat,A),fun(nat,A)),C3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),D3),C3))
          <=> sums(A,F3,D3) ) ) ) ).

% sums_mult2_iff
tff(fact_2751_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% le_floor_iff
tff(fact_2752_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_2753_powr__mono2_H,axiom,
    ! [A2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Yb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Yb,A2)),powr(real,Xb,A2)) ) ) ) ).

% powr_mono2'
tff(fact_2754_powr__less__mono2,axiom,
    ! [A2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Yb,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_2755_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Yb))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb))) ) ).

% le_floor_add
tff(fact_2756_powr__inj,axiom,
    ! [A2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,Xb) = powr(real,A2,Yb) )
        <=> ( Xb = Yb ) ) ) ) ).

% powr_inj
tff(fact_2757_gr__one__powr,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),powr(real,Xb,Yb)) ) ) ).

% gr_one_powr
tff(fact_2758_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,Xb)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),Xb)) ) ).

% int_add_floor
tff(fact_2759_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_2760_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_et(A,fun(fun(nat,A),fun(nat,A)),C3),F3),A2)
         => ( ( C3 != zero_zero(A) )
           => sums(A,F3,divide_divide(A,A2,C3)) ) ) ) ).

% sums_mult_D
tff(fact_2761_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S: A] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_eu(fun(nat,A),fun(nat,A),F3),S)
           => sums(A,F3,S) ) ) ) ).

% sums_Suc_imp
tff(fact_2762_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S: A] :
          ( sums(A,aTP_Lamp_eu(fun(nat,A),fun(nat,A),F3),S)
        <=> sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_2763_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_ev(fun(nat,A),fun(nat,A),F3),L)
         => sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_2764_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F3: fun(nat,A),S: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
             => ( aa(nat,A,F3,I3) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ew(nat,fun(fun(nat,A),fun(nat,A)),Nb),F3),S)
          <=> sums(A,F3,S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_2765_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A2: A,B2: A] : powr(A,Xb,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),powr(A,Xb,A2)),powr(A,Xb,B2)) ) ).

% powr_add
tff(fact_2766_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N3: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N3)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,N3)
               => ( aa(nat,A,F3,N2) = zero_zero(A) ) )
           => sums(A,F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),N3)) ) ) ) ).

% sums_finite
tff(fact_2767_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P2: fun(nat,$o),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P2))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ex(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P2),F3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(fun(nat,$o),set(nat),collect(nat),P2))) ) ) ).

% sums_If_finite
tff(fact_2768_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A5: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A5)
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ey(set(nat),fun(fun(nat,A),fun(nat,A)),A5),F3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),A5)) ) ) ).

% sums_If_finite_set
tff(fact_2769_prod__int__eq,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I2,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_bl(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_2770_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) ) ).

% one_add_floor
tff(fact_2771_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K2: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),Xb)) = K2 )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K2))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_2772_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Mb: nat,Z: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ez(nat,fun(A,fun(nat,A)),Mb),Z),aa(nat,A,power_power(A,Z),Mb)) ) ).

% powser_sums_if
tff(fact_2773_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_eo(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_2774_powr__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)) = aa(nat,real,power_power(real,Xb),Nb) ) ) ).

% powr_realpow
tff(fact_2775_powr__less__iff,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Yb)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),aa(real,real,log(B2),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_2776_less__powr__iff,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,Yb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xb)),Yb) ) ) ) ).

% less_powr_iff
tff(fact_2777_log__less__iff,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xb)),Yb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,Yb)) ) ) ) ).

% log_less_iff
tff(fact_2778_less__log__iff,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),aa(real,real,log(B2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Yb)),Xb) ) ) ) ).

% less_log_iff
tff(fact_2779_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),F3),Nb),S)
        <=> sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))) ) ) ).

% sums_iff_shift
tff(fact_2780_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S: A,Nb: nat] :
          ( sums(A,F3,S)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),F3),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),S),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_2781_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),F3),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),S),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),Nb))))
        <=> sums(A,F3,S) ) ) ).

% sums_iff_shift'
tff(fact_2782_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Nb)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_2783_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_2784_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G3: fun(nat,A),S2: A,A5: set(nat),S5: A,F3: fun(nat,A)] :
          ( sums(A,G3,S2)
         => ( aa(set(nat),$o,finite_finite2(nat),A5)
           => ( ( 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_fb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G3),F3)),A5)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fc(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G3),A5),F3),S5) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_2785_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),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_2786_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_bl(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)))) ).

% prod_int_plus_eq
tff(fact_2787_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
           => ( archim6421214686448440834_floor(A,Xb) = Z ) ) ) ) ).

% floor_unique
tff(fact_2788_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_2789_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P2: fun(int,$o),Ta: A] :
          ( aa(int,$o,P2,archim6421214686448440834_floor(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
             => aa(int,$o,P2,I4) ) ) ) ).

% floor_split
tff(fact_2790_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,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_2791_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb) ) ) ).

% less_floor_iff
tff(fact_2792_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_2793_binomial__maximum_H,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_2794_binomial__mono,axiom,
    ! [K2: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_mono
tff(fact_2795_binomial__antimono,axiom,
    ! [K2: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),K2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K2)) ) ) ) ).

% binomial_antimono
tff(fact_2796_binomial__maximum,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_2797_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb))),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_2798_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),one_one(real))) = divide_divide(real,one_one(real),Xb) ) ) ).

% powr_neg_one
tff(fact_2799_le__log__iff,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Yb),aa(real,real,log(B2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Yb)),Xb) ) ) ) ).

% le_log_iff
tff(fact_2800_log__le__iff,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xb)),Yb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Yb)) ) ) ) ).

% log_le_iff
tff(fact_2801_le__powr__iff,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Yb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xb)),Yb) ) ) ) ).

% le_powr_iff
tff(fact_2802_powr__le__iff,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Yb)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Yb),aa(real,real,log(B2),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_2803_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Nb)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_2804_ln__prod,axiom,
    ! [A: $tType,I6: set(A),F3: fun(A,real)] :
      ( aa(set(A),$o,finite_finite2(A),I6)
     => ( ! [I3: A] :
            ( member(A,I3,I6)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,I3)) )
       => ( aa(real,real,ln_ln(real),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),F3),I6)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_fd(fun(A,real),fun(A,real),F3)),I6) ) ) ) ).

% ln_prod
tff(fact_2805_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P,Q2)))),Q2)),P) ) ) ).

% floor_divide_lower
tff(fact_2806_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K2))) ) ).

% binomial_less_binomial_Suc
tff(fact_2807_binomial__strict__mono,axiom,
    ! [K2: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_strict_mono
tff(fact_2808_binomial__strict__antimono,axiom,
    ! [K2: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K2)) ) ) ) ).

% binomial_strict_antimono
tff(fact_2809_ln__powr__bound,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),divide_divide(real,powr(real,Xb,A2),A2)) ) ) ).

% ln_powr_bound
tff(fact_2810_ln__powr__bound2,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),Xb),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_2811_log__add__eq__powr,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),Xb)),Yb) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,Yb))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_2812_add__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Yb),aa(real,real,log(B2),Xb)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Yb)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_2813_minus__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Yb),aa(real,real,log(B2),Xb)) = aa(real,real,log(B2),divide_divide(real,powr(real,B2,Yb),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_2814_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),P),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,P,Q2)))),one_one(A))),Q2)) ) ) ).

% floor_divide_upper
tff(fact_2815_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => sums(A,power_power(A,C3),divide_divide(A,one_one(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C3))) ) ) ).

% geometric_sums
tff(fact_2816_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_round(A,Xb) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_2817_log__minus__eq__powr,axiom,
    ! [B2: real,Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(B2),Xb)),Yb) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,aa(real,real,uminus_uminus(real),Yb)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_2818_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P2: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L3))
           => ( ( ~ ( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L3,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P2,divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L3,aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P2,K),L3) ) )
       => aa(int,$o,aa(int,fun(int,$o),P2,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_2819_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Mb),Nb) = $ite(
        ( ( Mb = zero_zero(nat) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_2820_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Nb))) = divide_divide(real,one_one(real),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),Nb))) ) ) ).

% powr_neg_numeral
tff(fact_2821_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Mb),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_2822_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_2823_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),Nb)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),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),K2))) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_2824_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb) ) ).

% zero_less_binomial_iff
tff(fact_2825_binomial__n__0,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2826_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K2))) ).

% binomial_Suc_Suc
tff(fact_2827_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa(nat,nat,binomial(Nb),K2) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2) ) ).

% binomial_eq_0_iff
tff(fact_2828_binomial__0__Suc,axiom,
    ! [K2: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K2)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2829_binomial__1,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(nat,nat,suc,zero_zero(nat))) = Nb ).

% binomial_1
tff(fact_2830_binomial__eq__0,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2)
     => ( aa(nat,nat,binomial(Nb),K2) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2831_binomial__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
     => ( aa(nat,nat,binomial(Nb),K2) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)) ) ) ).

% binomial_symmetric
tff(fact_2832_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R2)),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R2)),K2)),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_2833_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),R2)),aa(nat,nat,power_power(nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_2834_zero__less__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K2)) ) ).

% zero_less_binomial
tff(fact_2835_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,suc,A2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),A2)) ).

% Suc_times_binomial_add
tff(fact_2836_choose__mult,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),Mb)),aa(nat,nat,binomial(Mb),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2))) ) ) ) ).

% choose_mult
tff(fact_2837_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Nb),aa(nat,A,semiring_1_of_nat(A),K2))),K2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K2))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2838_binomial__le__pow2,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% binomial_le_pow2
tff(fact_2839_choose__reduce__nat,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => ( aa(nat,nat,binomial(Nb),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_2840_times__binomial__minus1__eq,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,binomial(Nb),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2841_binomial__addition__formula,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,suc,K2))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K2)) ) ) ).

% binomial_addition_formula
tff(fact_2842_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P2: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I3: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),J2)
               => aa(int,$o,aa(int,fun(int,$o),P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P2,I3),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P2,A0),A1) ) ) ).

% upto.pinduct
tff(fact_2843_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fe(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ) ) ).

% choose_odd_sum
tff(fact_2844_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ff(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ) ) ).

% choose_even_sum
tff(fact_2845_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] : aa(A,A,arcosh(A),Xb) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),powr(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arcosh_def
tff(fact_2846_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_round(A,Xb) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,Xb)),archimedean_ceiling(A,Xb),archim6421214686448440834_floor(A,Xb)) ) ).

% round_altdef
tff(fact_2847_atMost__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,set(A),set_ord_atMost(A),Xb) = aa(A,set(A),set_ord_atMost(A),Yb) )
        <=> ( Xb = Yb ) ) ) ).

% atMost_eq_iff
tff(fact_2848_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( member(A,I2,aa(A,set(A),set_ord_atMost(A),K2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),K2) ) ) ).

% atMost_iff
tff(fact_2849_finite__atMost,axiom,
    ! [K2: nat] : aa(set(nat),$o,finite_finite2(nat),aa(nat,set(nat),set_ord_atMost(nat),K2)) ).

% finite_atMost
tff(fact_2850_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),Xb)),aa(A,set(A),set_ord_atMost(A),Yb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ).

% atMost_subset_iff
tff(fact_2851_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] :
          ( ( real_Vector_of_real(A,Xb) = zero_zero(A) )
        <=> ( Xb = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_2852_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_2853_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Yb: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Yb)) ) ).

% of_real_add
tff(fact_2854_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_2855_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),H),H2) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_2856_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = 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),G3),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) ) ).

% sum.atMost_Suc
tff(fact_2857_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) ) ).

% prod.atMost_Suc
tff(fact_2858_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_2859_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_2860_infinite__Iic,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A2: A] : ~ aa(set(A),$o,finite_finite2(A),aa(A,set(A),set_ord_atMost(A),A2)) ) ).

% infinite_Iic
tff(fact_2861_not__Iic__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H2: A,L: A,H: A] : aa(A,set(A),set_ord_atMost(A),H2) != set_or1337092689740270186AtMost(A,L,H) ) ).

% not_Iic_eq_Icc
tff(fact_2862_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fg(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_2863_atMost__atLeast0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_atMost(nat),Nb) = set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb) ).

% atMost_atLeast0
tff(fact_2864_lessThan__Suc__atMost,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K2)) = aa(nat,set(nat),set_ord_atMost(nat),K2) ).

% lessThan_Suc_atMost
tff(fact_2865_atMost__Suc,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,K2)),aa(nat,set(nat),set_ord_atMost(nat),K2)) ).

% atMost_Suc
tff(fact_2866_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A,H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_or1337092689740270186AtMost(A,L2,H2)) ) ).

% not_Iic_le_Icc
tff(fact_2867_finite__nat__iff__bounded__le,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_atMost(nat),K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_2868_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,Xb)) ) ).

% frac_ge_0
tff(fact_2869_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,Xb)),one_one(A)) ) ).

% frac_lt_1
tff(fact_2870_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) = archimedean_frac(A,Xb) ) ).

% frac_1_eq
tff(fact_2871_atMost__nat__numeral,axiom,
    ! [K2: num] : aa(nat,set(nat),set_ord_atMost(nat),aa(num,nat,numeral_numeral(nat),K2)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K2))) ).

% atMost_nat_numeral
tff(fact_2872_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A2)),aa(A,set(A),set_ord_lessThan(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_2873_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,Xb))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_2874_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ci(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% sum.atMost_Suc_shift
tff(fact_2875_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),I2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dg(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_atMost(nat),I2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,aa(nat,nat,suc,I2))) ) ).

% sum_telescope
tff(fact_2876_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Nb: nat,D3: fun(nat,A)] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C3),X4)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),D3),X4)),aa(nat,set(nat),set_ord_atMost(nat),Nb))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
             => ( aa(nat,A,C3,I4) = aa(nat,A,D3,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_2877_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ec(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% prod.atMost_Suc_shift
tff(fact_2878_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fi(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fk(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.nested_swap'
tff(fact_2879_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.nested_swap'
tff(fact_2880_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fo(nat,fun(nat,nat),R2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_2881_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fp(nat,fun(nat,nat),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)),one_one(nat))),Mb) ).

% choose_rising_sum(2)
tff(fact_2882_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_fp(nat,fun(nat,nat),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_2883_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( archimedean_frac(A,Xb) = Xb )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_2884_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Nb: nat] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C3),X4)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
             => ( aa(nat,A,C3,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_2885_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C3: fun(nat,A),Nb: nat,K2: nat] :
          ( ! [W: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fq(fun(nat,A),fun(A,fun(nat,A)),C3),W)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => ( aa(nat,A,C3,K2) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_2886_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Yb))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Yb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Yb))),one_one(A))) ) ).

% frac_add
tff(fact_2887_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ci(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% sum.atMost_shift
tff(fact_2888_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_atMost(nat),Mb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)))) ) ).

% sum_up_index_split
tff(fact_2889_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ec(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% prod.atMost_shift
tff(fact_2890_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atMost(nat),Nb)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_2891_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_fr(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ft(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% sum.triangle_reindex_eq
tff(fact_2892_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_fr(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fv(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% prod.triangle_reindex_eq
tff(fact_2893_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fw(nat,fun(nat,fun(nat,nat)),Mb),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_2894_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fx(nat,fun(nat,fun(nat,fun(nat,nat))),Mb),Nb),R2)),aa(nat,set(nat),set_ord_atMost(nat),R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),R2) ).

% vandermonde
tff(fact_2895_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,Xb),aa(nat,nat,suc,Nb))) ) ).

% sum_gp_basic
tff(fact_2896_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Nb: nat] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_fy(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
              & ( aa(nat,A,C3,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_2897_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,Nb: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_fy(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_2898_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: fun(nat,A),A2: A,Nb: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),C3),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) )
         => ~ ! [B4: fun(nat,A)] :
                ~ ! [Z3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),C3),Z3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z3),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),B4),Z3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ).

% polyfun_linear_factor_root
tff(fact_2899_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: fun(nat,A),Nb: nat,A2: A] :
        ? [B4: fun(nat,A)] :
        ! [Z3: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),C3),Z3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z3),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),B4),Z3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),C3),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% polyfun_linear_factor
tff(fact_2900_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Mb: nat,Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Xb),Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ).

% sum_power_shift
tff(fact_2901_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ga(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ft(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.triangle_reindex
tff(fact_2902_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ga(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fv(fun(nat,fun(nat,A)),fun(nat,A),G3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.triangle_reindex
tff(fact_2903_binomial,axiom,
    ! [A2: nat,B2: nat,Nb: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Nb) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_gb(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ).

% binomial
tff(fact_2904_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cs(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% sum.in_pairs_0
tff(fact_2905_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Mb: nat,A2: fun(nat,A),Nb: nat,B2: fun(nat,A),Xb: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I3)
             => ( aa(nat,A,A2,I3) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),J2)
               => ( aa(nat,A,B2,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Mb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),B2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = 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_gd(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ) ) ) ).

% polynomial_product
tff(fact_2906_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ek(fun(nat,A),fun(nat,A),G3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% prod.in_pairs_0
tff(fact_2907_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Nb: nat,K2: A] :
          ( ! [X4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C3),X4)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = K2
        <=> ( ( aa(nat,A,C3,zero_zero(nat)) = K2 )
            & ! [X4: nat] :
                ( member(nat,X4,set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C3,X4) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_2908_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,Nb: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Nb) = 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_ge(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% binomial_ring
tff(fact_2909_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),Nb) = 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_gf(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% pochhammer_binomial_sum
tff(fact_2910_polynomial__product__nat,axiom,
    ! [Mb: nat,A2: fun(nat,nat),Nb: nat,B2: fun(nat,nat),Xb: nat] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I3)
         => ( aa(nat,nat,A2,I3) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),J2)
           => ( aa(nat,nat,B2,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_gg(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Mb))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_gg(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = 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_gi(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_2911_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Yb))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Yb)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Yb))),one_one(int))) ) ).

% floor_add
tff(fact_2912_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),P)
           => ( 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_gj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),P)) = 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_gk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2913_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),P)
           => ( 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_gl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),P)) = 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_gm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_2914_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,Z: A,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( ( aa(nat,A,power_power(A,Z),Nb) = 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_gn(nat,fun(A,fun(A,fun(nat,A))),Nb),Z),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_2915_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,Xb),aa(nat,nat,suc,Nb))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp0
tff(fact_2916_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( ( Nb != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_go(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2917_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Yb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),A2),Yb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)),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_gq(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Yb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_2918_binomial__r__part__sum,axiom,
    ! [Mb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb)) ).

% binomial_r_part_sum
tff(fact_2919_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gr(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_2920_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E3: real,C3: fun(nat,A),Nb: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => ? [M7: real] :
            ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M7),real_V7770717601297561774m_norm(A,Z3))
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),C3),Z3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)))),aa(real,real,aa(real,fun(real,real),times_times(real),E3),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_2921_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Yb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),A2),Yb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)),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_gu(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Yb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ) ).

% polyfun_diff
tff(fact_2922_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] : arsinh(A,Xb) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arsinh_def
tff(fact_2923_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gv(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Mb)),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_2924_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Mb))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb)) ) ).

% gbinomial_r_part_sum
tff(fact_2925_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gv(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Mb)) = 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,Mb)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,Mb))) ) ).

% gchoose_row_sum_weighted
tff(fact_2926_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),Xb)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_2927_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),exp(real,one_one(real))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq1
tff(fact_2928_of__nat__id,axiom,
    ! [Nb: nat] : aa(nat,nat,semiring_1_of_nat(nat),Nb) = Nb ).

% of_nat_id
tff(fact_2929_exp__less__cancel__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Yb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ).

% exp_less_cancel_iff
tff(fact_2930_exp__less__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Yb)) ) ).

% exp_less_mono
tff(fact_2931_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_2932_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_2933_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K2: nat] : aa(nat,A,gbinomial(A,zero_zero(A)),aa(nat,nat,suc,K2)) = zero_zero(A) ) ).

% gbinomial_0(2)
tff(fact_2934_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_2935_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% gbinomial_Suc0
tff(fact_2936_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% one_less_exp_iff
tff(fact_2937_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_2938_exp__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( exp(real,aa(real,real,ln_ln(real),Xb)) = Xb ) ) ).

% exp_ln
tff(fact_2939_exp__ln__iff,axiom,
    ! [Xb: real] :
      ( ( exp(real,aa(real,real,ln_ln(real),Xb)) = Xb )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% exp_ln_iff
tff(fact_2940_exp__less__cancel,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Yb))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ).

% exp_less_cancel
tff(fact_2941_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : exp(A,Xb) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_2942_exp__total,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
     => ? [X3: real] : exp(real,X3) = Yb ) ).

% exp_total
tff(fact_2943_exp__gt__zero,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),exp(real,Xb)) ).

% exp_gt_zero
tff(fact_2944_not__exp__less__zero,axiom,
    ! [Xb: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_2945_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb) = aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Xb) )
         => ( exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xb)),exp(A,Yb)) ) ) ) ).

% exp_add_commuting
tff(fact_2946_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xb)),exp(A,Yb)) = exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) ) ).

% mult_exp_exp
tff(fact_2947_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xb)) ) ).

% exp_gt_one
tff(fact_2948_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A2),K2)),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2))) ) ).

% gbinomial_Suc_Suc
tff(fact_2949_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K2) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_2950_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I6: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( exp(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),I6)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aTP_Lamp_gw(fun(A,B),fun(A,B),F3)),I6) ) ) ) ).

% exp_sum
tff(fact_2951_ln__ge__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Yb),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Yb)),Xb) ) ) ).

% ln_ge_iff
tff(fact_2952_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K2)) ) ).

% gbinomial_addition_formula
tff(fact_2953_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K2: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K2)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),K2))),K2)),aa(nat,A,gbinomial(A,A2),K2)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_2954_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,A2),K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K2)),aa(nat,A,gbinomial(A,A2),K2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_mult_1
tff(fact_2955_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K2)),aa(nat,A,gbinomial(A,A2),K2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_mult_1'
tff(fact_2956_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A,A2: A] :
          powr(A,Xb,A2) = $ite(Xb = zero_zero(A),zero_zero(A),exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),Xb)))) ) ).

% powr_def
tff(fact_2957_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,gbinomial(A,A2),K2)) ) ).

% Suc_times_gbinomial
tff(fact_2958_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Mb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),Mb)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Mb)),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2959_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(nat,A,power_power(A,exp(A,divide_divide(A,Xb,aa(nat,A,semiring_1_of_nat(A),Nb)))),Nb) = exp(A,Xb) ) ) ) ).

% exp_divide_power_eq
tff(fact_2960_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tanh(A),Xb) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% tanh_altdef
tff(fact_2961_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gx(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),Nb))),one_one(A))),Nb) ) ).

% gbinomial_parallel_sum
tff(fact_2962_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_rec
tff(fact_2963_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2)))),aa(nat,A,gbinomial(A,A2),K2)) ) ).

% gbinomial_factors
tff(fact_2964_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A2)),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K2)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A))),K2)) ) ).

% gbinomial_minus
tff(fact_2965_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
         => ( aa(nat,A,gbinomial(A,A2),K2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K2)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2966_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,A2: A,Xb: A,Yb: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_gy(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Yb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_gz(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Yb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) ) ).

% gbinomial_partial_sum_poly
tff(fact_2967_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ha(nat,fun(nat,A),K2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2968_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Nb))),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_2969_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),divide_divide(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,aa(real,real,uminus_uminus(real),Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_2970_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
         => ( aa(nat,A,gbinomial(A,A2),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_2971_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hb(nat,fun(nat,A),Mb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb) ) ).

% gbinomial_sum_nat_pow2
tff(fact_2972_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,A2: A,Xb: A,Yb: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_gy(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Yb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hc(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Yb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_2973_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cot(real),Xb)) ) ) ).

% cot_gt_zero
tff(fact_2974_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),exp(real,one_one(real)))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq2
tff(fact_2975_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] :
          aa(nat,A,gbinomial(A,A2),K2) = $ite(K2 = zero_zero(nat),one_one(A),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_hd(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K2))) ) ).

% gbinomial_code
tff(fact_2976_sum__pos__lt__pair,axiom,
    ! [F3: fun(nat,real),K2: nat] :
      ( summable(real,F3)
     => ( ! [D2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)))),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),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)))),D2)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F3),aa(nat,set(nat),set_ord_lessThan(nat),K2))),suminf(real,F3)) ) ) ).

% sum_pos_lt_pair
tff(fact_2977_binomial__code,axiom,
    ! [Nb: nat,K2: nat] :
      aa(nat,nat,binomial(Nb),K2) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)),divide_divide(nat,set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)),one_one(nat)),Nb,one_one(nat)),semiring_char_0_fact(nat,K2))) ) ).

% binomial_code
tff(fact_2978_modulo__int__unfold,axiom,
    ! [K2: int,Mb: nat,L: int,Nb: nat] :
      modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K2)),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K2) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K2)),aa(nat,int,semiring_1_of_nat(int),Mb)),
        $ite(sgn_sgn(int,K2) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))))) ) ).

% modulo_int_unfold
tff(fact_2979_powr__int,axiom,
    ! [Xb: real,I2: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),I2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2),aa(nat,real,power_power(real,Xb),aa(int,nat,nat2,I2)),divide_divide(real,one_one(real),aa(nat,real,power_power(real,Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I2))))) ) ) ).

% powr_int
tff(fact_2980_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% sgn_sgn
tff(fact_2981_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_2982_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_2983_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_2984_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,divide_divide(A,A2,B2)) = divide_divide(A,sgn_sgn(A,A2),sgn_sgn(A,B2)) ) ).

% sgn_divide
tff(fact_2985_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),sgn_sgn(A,A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_2986_nat__int,axiom,
    ! [Nb: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),Nb)) = Nb ).

% nat_int
tff(fact_2987_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_en(nat,A)) ) ).

% summable_zero
tff(fact_2988_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I2: nat,F3: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ep(nat,fun(fun(nat,A),fun(nat,A)),I2),F3)) ) ).

% summable_single
tff(fact_2989_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),F3),K2))
        <=> summable(A,F3) ) ) ).

% summable_iff_shift
tff(fact_2990_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),sgn_sgn(A,A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_2991_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),sgn_sgn(A,A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_2992_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : divide_divide(A,A2,sgn_sgn(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),sgn_sgn(A,B2)) ) ).

% divide_sgn
tff(fact_2993_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2994_nat__numeral,axiom,
    ! [K2: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K2)) = aa(num,nat,numeral_numeral(nat),K2) ).

% nat_numeral
tff(fact_2995_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_et(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_cmult_iff
tff(fact_2996_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_he(fun(nat,A),fun(A,fun(nat,A)),F3),C3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_divide_iff
tff(fact_2997_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A5: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A5)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ey(set(nat),fun(fun(nat,A),fun(nat,A)),A5),F3)) ) ) ).

% summable_If_finite_set
tff(fact_2998_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P2: fun(nat,$o),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P2))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ex(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P2),F3)) ) ) ).

% summable_If_finite
tff(fact_2999_nat__of__bool,axiom,
    ! [P2: $o] : aa(int,nat,nat2,aa($o,int,zero_neq_one_of_bool(int),(P2))) = aa($o,nat,zero_neq_one_of_bool(nat),(P2)) ).

% nat_of_bool
tff(fact_3000_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_3001_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_3002_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_3003_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_3004_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_3005_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa(int,nat,nat2,I2) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_3006_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% zless_nat_conj
tff(fact_3007_nat__neg__numeral,axiom,
    ! [K2: num] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_3008_nat__zminus__int,axiom,
    ! [Nb: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_3009_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_3010_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R2)),L)),K2)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K2)
        & ( ( R2 = zero_zero(int) )
         => ( K2 = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3011_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),sgn_sgn(int,R2))),K2)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K2)
        & ( ( R2 = zero_zero(int) )
         => ( K2 = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3012_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R2)),K2))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K2)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3013_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),K2),sgn_sgn(int,R2)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K2)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3014_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_3015_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_3016_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_3017_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : sgn_sgn(A,aa(nat,A,semiring_1_of_nat(A),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% sgn_of_nat
tff(fact_3018_diff__nat__numeral,axiom,
    ! [V: num,V4: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V4)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V4))) ).

% diff_nat_numeral
tff(fact_3019_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa(int,nat,nat2,Yb) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xb)),Nb) )
    <=> ( Yb = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_3020_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xb)),Nb) = aa(int,nat,nat2,Yb) )
    <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_3021_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( summable(A,power_power(A,C3))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_3022_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,Xb))),A2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_3023_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_3024_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_3025_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xb)),Nb)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_3026_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,A2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xb)),Nb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_3027_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,A2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xb)),Nb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_3028_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xb)),Nb)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_3029_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_3030_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_3031_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( ( sgn_sgn(A,Xb) = zero_zero(A) )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_3032_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,B2)) ) ).

% sgn_mult
tff(fact_3033_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( sgn_sgn(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = sgn_sgn(A,A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_3034_fact__ge__self,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_self
tff(fact_3035_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,Mb)),semiring_char_0_fact(nat,Nb)) ) ).

% fact_mono_nat
tff(fact_3036_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_3037_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C3: A] :
          ( summable(A,aTP_Lamp_hf(A,fun(nat,A),C3))
        <=> ( C3 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_3038_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),aa(nat,real,G3,N2)) )
         => ( summable(real,G3)
           => summable(A,F3) ) ) ) ).

% summable_comparison_test
tff(fact_3039_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G3: fun(nat,real),N3: nat,F3: fun(nat,A)] :
          ( summable(real,G3)
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N2)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),aa(nat,real,G3,N2)) )
           => summable(A,F3) ) ) ) ).

% summable_comparison_test'
tff(fact_3040_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( summable(A,F3)
         => ( summable(A,G3)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3)) ) ) ) ).

% summable_add
tff(fact_3041_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( summable(A,F3)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),F3),K2)) ) ) ).

% summable_ignore_initial_segment
tff(fact_3042_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),F3),Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,real)),F3),Z)) ) ) ) ).

% powser_insidea
tff(fact_3043_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,G3,N2))
         => ( summable(A,F3)
           => ( summable(A,G3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F3)),suminf(A,G3)) ) ) ) ) ).

% suminf_le
tff(fact_3044_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N3: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N3)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,N3)
               => ( aa(nat,A,F3,N2) = zero_zero(A) ) )
           => summable(A,F3) ) ) ) ).

% summable_finite
tff(fact_3045_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,Mb)),semiring_char_0_fact(nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_3046_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) != sgn_sgn(A,A2) )
         => ( ( sgn_sgn(A,A2) != zero_zero(A) )
           => ( ( sgn_sgn(A,B2) != zero_zero(A) )
             => ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),sgn_sgn(A,B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_3047_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_3048_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_3049_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_zero
tff(fact_3050_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Nb)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_3051_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_gt_zero
tff(fact_3052_nat__mono,axiom,
    ! [Xb: int,Yb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Yb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Yb)) ) ).

% nat_mono
tff(fact_3053_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N2: nat,L3: int] : K2 != aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L3)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_sgnE
tff(fact_3054_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_3055_all__nat,axiom,
    ! [P2: fun(nat,$o)] :
      ( ! [X_13: nat] : aa(nat,$o,P2,X_13)
    <=> ! [X4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X4)
         => aa(nat,$o,P2,aa(int,nat,nat2,X4)) ) ) ).

% all_nat
tff(fact_3056_ex__nat,axiom,
    ! [P2: fun(nat,$o)] :
      ( ? [X_13: nat] : aa(nat,$o,P2,X_13)
    <=> ? [X4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X4)
          & aa(nat,$o,P2,aa(int,nat,nat2,X4)) ) ) ).

% ex_nat
tff(fact_3057_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_1
tff(fact_3058_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Mb)),semiring_char_0_fact(A,Nb)) ) ) ).

% fact_mono
tff(fact_3059_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_et(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
         => ( ( C3 != zero_zero(A) )
           => summable(A,F3) ) ) ) ).

% summable_mult_D
tff(fact_3060_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,Mb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,Mb)) ) ) ).

% fact_dvd
tff(fact_3061_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,power_power(A,zero_zero(A))) ) ).

% summable_zero_power
tff(fact_3062_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( summable(A,F3)
         => ( summable(A,G3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F3)),suminf(A,G3)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3)) ) ) ) ) ).

% suminf_add
tff(fact_3063_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2))
           => ( ( suminf(A,F3) = zero_zero(A) )
            <=> ! [N: nat] : aa(nat,A,F3,N) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3064_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F3)) ) ) ) ).

% suminf_nonneg
tff(fact_3065_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,N2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3)) ) ) ) ).

% suminf_pos
tff(fact_3066_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_3067_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_3068_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),semiring_char_0_fact(nat,Nb)) ) ) ).

% dvd_fact
tff(fact_3069_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_hh(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_zero_power'
tff(fact_3070_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_eo(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_0_powser
tff(fact_3071_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% nat_mono_iff
tff(fact_3072_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R2)))) ) ).

% of_nat_ceiling
tff(fact_3073_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Mb)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_3074_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xb)),Nb)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_le_iff
tff(fact_3075_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_3076_int__eq__iff,axiom,
    ! [Mb: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Mb) = Z )
    <=> ( ( Mb = aa(int,nat,nat2,Z) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_3077_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Mb: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hi(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F3),Mb),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3078_nat__int__add,axiom,
    ! [A2: nat,B2: nat] : aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2) ).

% nat_int_add
tff(fact_3079_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Mb)),semiring_char_0_fact(A,Nb)) ) ) ) ).

% fact_less_mono
tff(fact_3080_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: nat,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,Nb))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb))) ) ).

% fact_fact_dvd_fact
tff(fact_3081_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( modulo_modulo(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,Mb)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3082_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Nb)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,Nb),Nb))) ) ).

% fact_le_power
tff(fact_3083_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),aa(nat,real,G3,N2)) )
         => ( summable(real,G3)
           => summable(real,aTP_Lamp_hj(fun(nat,A),fun(nat,real),F3)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3084_nat__plus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_3085_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I2: nat] :
          ( summable(A,F3)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3)) ) ) ) ) ).

% suminf_pos2
tff(fact_3086_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3))
            <=> ? [I4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I4)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3087_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Xb: A] :
          ( summable(A,F3)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),N2))),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F3)),Xb) ) ) ) ).

% suminf_le_const
tff(fact_3088_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_3089_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_3090_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          sgn_sgn(A,Xb) = $ite(
            Xb = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_3091_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R2)))),R2) ) ) ).

% of_nat_floor
tff(fact_3092_zsgn__def,axiom,
    ! [I2: int] :
      sgn_sgn(int,I2) = $ite(
        I2 = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I2),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_3093_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,semiring_char_0_fact(nat,Nb),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),R2)))),aa(nat,nat,power_power(nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_3094_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hk(fun(nat,A),fun(A,fun(nat,A)),F3),Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_hk(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ) ).

% powser_inside
tff(fact_3095_binomial__fact__lemma,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))),aa(nat,nat,binomial(Nb),K2)) = semiring_char_0_fact(nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_3096_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_3097_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Xb: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2))
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),N2))),Xb)
           => summable(A,F3) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3098_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          real_V7770717601297561774m_norm(A,sgn_sgn(A,Xb)) = $ite(Xb = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_3099_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),W2)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_3100_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B5: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N2))
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),aa(nat,set(nat),set_ord_atMost(nat),N2))),B5)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_3101_nat__eq__iff,axiom,
    ! [W2: int,Mb: nat] :
      ( ( aa(int,nat,nat2,W2) = Mb )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2),W2 = aa(nat,int,semiring_1_of_nat(int),Mb),Mb = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_3102_nat__eq__iff2,axiom,
    ! [Mb: nat,W2: int] :
      ( ( Mb = aa(int,nat,nat2,W2) )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2),W2 = aa(nat,int,semiring_1_of_nat(int),Mb),Mb = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_3103_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => summable(A,power_power(A,Xb)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3104_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => summable(A,power_power(A,C3)) ) ) ).

% summable_geometric
tff(fact_3105_split__nat,axiom,
    ! [P2: fun(nat,$o),I2: int] :
      ( aa(nat,$o,P2,aa(int,nat,nat2,I2))
    <=> ( ! [N: nat] :
            ( ( I2 = aa(nat,int,semiring_1_of_nat(int),N) )
           => aa(nat,$o,P2,N) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),zero_zero(int))
         => aa(nat,$o,P2,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_3106_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,archim6421214686448440834_floor(A,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_3107_le__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(int,nat,nat2,K2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K2) ) ) ).

% le_nat_iff
tff(fact_3108_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( suminf(A,aTP_Lamp_eu(fun(nat,A),fun(nat,A),F3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F3)),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3109_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_3110_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_3111_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z4),Z)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_3112_nat__diff__distrib_H,axiom,
    ! [Xb: int,Yb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Yb)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Yb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Yb)) ) ) ) ).

% nat_diff_distrib'
tff(fact_3113_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(nat,int,power_power(int,Z),Nb)) = aa(nat,nat,power_power(nat,aa(int,nat,nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_3114_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xb)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_3115_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))),semiring_char_0_fact(A,Nb)) ) ) ).

% choose_dvd
tff(fact_3116_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_3117_le__nat__floor,axiom,
    ! [Xb: nat,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xb)),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(int,nat,nat2,archim6421214686448440834_floor(real,A2))) ) ).

% le_nat_floor
tff(fact_3118_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( semiring_char_0_fact(nat,Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Nb)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ca(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_3119_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I6: set(nat)] :
          ( summable(A,F3)
         => ( aa(set(nat),$o,finite_finite2(nat),I6)
           => ( ! [N2: nat] :
                  ( member(nat,N2,aa(set(nat),set(nat),uminus_uminus(set(nat)),I6))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),I6)),suminf(A,F3)) ) ) ) ) ).

% sum_le_suminf
tff(fact_3120_nat__2,axiom,
    aa(int,nat,nat2,aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_3121_binomial__altdef__nat,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
     => ( aa(nat,nat,binomial(Nb),K2) = divide_divide(nat,semiring_char_0_fact(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_3122_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_3123_nat__less__iff,axiom,
    ! [W2: int,Mb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),Mb)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),Mb)) ) ) ).

% nat_less_iff
tff(fact_3124_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_3125_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( summable(A,F3)
         => ( suminf(A,F3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),F3),K2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),K2))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_3126_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( summable(A,F3)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),F3),K2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F3)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),K2))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_3127_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_3128_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( divide_divide(nat,semiring_char_0_fact(nat,Mb),semiring_char_0_fact(nat,Nb)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ca(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),Mb)) ) ) ).

% fact_div_fact
tff(fact_3129_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),zero_zero(int)),
        aa(int,nat,nat2,Z),
        $let(
          d: int,
          d:= aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z4),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),d),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_3130_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Nb: nat] :
          ( summable(A,F3)
         => ( ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,M)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),suminf(A,F3)) ) ) ) ).

% sum_less_suminf
tff(fact_3131_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hk(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_hk(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F3,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3132_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hk(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),Z) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hk(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3133_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),E3: real] :
          ( summable(A,F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ~ ! [N8: nat] :
                  ~ ! [M3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),M3)
                     => ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,M3,N4)))),E3) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3134_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( summable(A,F3)
           => ? [N8: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N4)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),F3),N4)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_3135_summable__power__series,axiom,
    ! [F3: fun(nat,real),Z: real] :
      ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,I3)),one_one(real))
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,I3))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Z)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_hm(fun(nat,real),fun(real,fun(nat,real)),F3),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3136_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A2: fun(nat,A),M4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),R0)
           => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N2))),aa(nat,real,power_power(real,R0),N2))),M4)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_hn(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_3137_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: int] :
          aa(int,A,ring_1_of_int(A),K2) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K2)))),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K2))) ) ).

% of_int_of_nat
tff(fact_3138_nat__dvd__iff,axiom,
    ! [Z: int,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(int,nat,nat2,Z)),Mb)
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),aa(nat,int,semiring_1_of_nat(int),Mb)),Mb = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_3139_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Mb: nat] :
          semiring_char_0_fact(A,Mb) = $ite(Mb = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_3140_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_3141_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K2))) = divide_divide(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2))) ) ) ) ).

% fact_binomial
tff(fact_3142_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K2)) = divide_divide(A,semiring_char_0_fact(A,Nb),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))) ) ) ) ).

% binomial_fact
tff(fact_3143_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C3: real,N3: nat,F3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),one_one(real))
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N2)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,aa(nat,nat,suc,N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),C3),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2)))) )
           => summable(A,F3) ) ) ) ).

% summable_ratio_test
tff(fact_3144_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Nb: nat,I2: nat] :
          ( summable(A,F3)
         => ( ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,M)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),I2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),suminf(A,F3)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3145_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),K2) = divide_divide(A,comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K2))),one_one(A)),K2),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_pochhammer'
tff(fact_3146_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: real,Nb: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xb = zero_zero(real) )
         => ( ( Nb != zero_zero(nat) )
           => ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ho(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xb),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3147_Maclaurin__lemma,axiom,
    ! [H: real,F3: fun(real,real),J: fun(nat,real),Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ? [B8: real] : aa(real,real,F3,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_hp(real,fun(fun(nat,real),fun(nat,real)),H),J)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),B8),divide_divide(real,aa(nat,real,power_power(real,H),Nb),semiring_char_0_fact(real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_3148_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2)) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hq(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K2)),semiring_char_0_fact(A,aa(nat,nat,suc,K2))) ) ).

% gbinomial_Suc
tff(fact_3149_divide__int__unfold,axiom,
    ! [K2: int,Mb: nat,L: int,Nb: nat] :
      divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K2)),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K2) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(sgn_sgn(int,K2) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Mb,Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,Mb,Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_3150_sin__coeff__def,axiom,
    ! [X: nat] :
      sin_coeff(X) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X),zero_zero(real),divide_divide(real,aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),semiring_char_0_fact(real,X))) ).

% sin_coeff_def
tff(fact_3151_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => summable(real,aTP_Lamp_hr(real,fun(nat,real),Xb)) ) ).

% summable_arctan_series
tff(fact_3152_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C3: fun(nat,A),Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hs(fun(nat,A),fun(A,fun(nat,A)),C3),Xb))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_ht(fun(nat,A),fun(A,fun(nat,A)),C3),Xb),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hs(fun(nat,A),fun(A,fun(nat,A)),C3),Xb))) ) ) ).

% diffs_equiv
tff(fact_3153_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T5))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xb))
            & ( exp(real,Xb) = 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_hu(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3154_sin__paired,axiom,
    ! [Xb: real] : sums(real,aTP_Lamp_hv(real,fun(nat,real),Xb),sin(real,Xb)) ).

% sin_paired
tff(fact_3155_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_3156_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_abs
tff(fact_3157_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_3158_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_3159_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_3160_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_3161_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),A2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% abs_mult_self_eq
tff(fact_3162_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_3163_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_3164_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: 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)) ) ).

% abs_divide
tff(fact_3165_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_3166_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus
tff(fact_3167_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: A,K2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Mb),aa(A,A,abs_abs(A),K2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Mb),K2) ) ) ).

% dvd_abs_iff
tff(fact_3168_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: A,K2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,abs_abs(A),Mb)),K2)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Mb),K2) ) ) ).

% abs_dvd_iff
tff(fact_3169_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_3170_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ).

% abs_of_nat
tff(fact_3171_of__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int] : aa(int,A,ring_1_of_int(A),aa(int,int,abs_abs(int),Xb)) = aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Xb)) ) ).

% of_int_abs
tff(fact_3172_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: $o] : aa(A,A,abs_abs(A),aa($o,A,zero_neq_one_of_bool(A),(P2))) = aa($o,A,zero_neq_one_of_bool(A),(P2)) ) ).

% abs_bool_eq
tff(fact_3173_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_3174_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% abs_le_self_iff
tff(fact_3175_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_3176_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A2))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_3177_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3178_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_hw(fun(B,A),fun(B,A),F3)),A5)) ) ).

% sum_abs
tff(fact_3179_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_3180_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_3181_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_3182_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_3183_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),Xb)) ) ) ).

% artanh_minus_real
tff(fact_3184_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,abs_abs(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_3185_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_3186_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_3187_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_hw(fun(B,A),fun(B,A),F3)),A5)) ) ).

% sum_abs_ge_zero
tff(fact_3188_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),Nb))
        <=> ( ( A2 != zero_zero(A) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3189_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_3190_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: real,B2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_3191_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,abs_abs(A),Xb) = aa(A,A,abs_abs(A),Yb) )
        <=> ( ( Xb = Yb )
            | ( Xb = aa(A,A,uminus_uminus(A),Yb) ) ) ) ) ).

% abs_eq_iff
tff(fact_3192_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_3193_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_3194_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_3195_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_self
tff(fact_3196_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% abs_le_D1
tff(fact_3197_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K2: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K2) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),L),K2) ) ) ).

% dvd_if_abs_eq
tff(fact_3198_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_3199_sin__zero__pi__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),pi)
     => ( ( sin(real,Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3200_sgn__power__injE,axiom,
    ! [A2: real,Nb: nat,Xb: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,A2)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),A2)),Nb)) = Xb )
     => ( ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,B2)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),B2)),Nb)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_3201_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_zero
tff(fact_3202_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_3203_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_3204_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_3205_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D3)) ) ) ) ).

% abs_mult_less
tff(fact_3206_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),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_3207_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),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_3208_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),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_3209_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_3210_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_minus_self
tff(fact_3211_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_le_iff
tff(fact_3212_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ).

% abs_le_D2
tff(fact_3213_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2) ) ) ) ).

% abs_leI
tff(fact_3214_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_less_iff
tff(fact_3215_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K2: A] : aa(A,A,abs_abs(A),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),K2),sgn_sgn(A,K2)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_3216_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),sgn_sgn(A,A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_3217_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_3218_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,Xb)),aa(A,A,abs_abs(A),Xb)) = Xb ) ).

% mult_sgn_abs
tff(fact_3219_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_3220_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xb: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),E2) )
         => ( Xb = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_3221_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Yb)),Xb) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Xb)) ) ) ) ).

% abs_mult_pos
tff(fact_3222_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_3223_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_3224_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_3225_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A2))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_3226_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),Nb)) ) ).

% zero_le_power_abs
tff(fact_3227_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Yb)
         => ( divide_divide(A,aa(A,A,abs_abs(A),Xb),Yb) = aa(A,A,abs_abs(A),divide_divide(A,Xb,Yb)) ) ) ) ).

% abs_div_pos
tff(fact_3228_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),A2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)),aa(A,A,uminus_uminus(A),A2),A2) ) ).

% abs_if
tff(fact_3229_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_3230_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X: A] :
          aa(A,A,abs_abs(A),X) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),zero_zero(A)),aa(A,A,uminus_uminus(A),X),X) ) ).

% abs_if_raw
tff(fact_3231_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_3232_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),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_3233_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C3: A,D3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D3)))),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),C3))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)))) ) ).

% abs_diff_triangle_ineq
tff(fact_3234_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_3235_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_3236_sin__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero
tff(fact_3237_abs__real__def,axiom,
    ! [A2: real] :
      aa(real,real,abs_abs(real),A2) = $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real)),aa(real,real,uminus_uminus(real),A2),A2) ).

% abs_real_def
tff(fact_3238_summable__rabs__comparison__test,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real)] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F3,N2))),aa(nat,real,G3,N2)) )
     => ( summable(real,G3)
       => summable(real,aTP_Lamp_hx(fun(nat,real),fun(nat,real),F3)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3239_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),Xb))) ) ).

% abs_add_one_gt_zero
tff(fact_3240_sgn__real__def,axiom,
    ! [A2: real] :
      sgn_sgn(real,A2) = $ite(
        A2 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_3241_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% of_int_leD
tff(fact_3242_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ) ).

% of_int_lessD
tff(fact_3243_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),pi)
       => ( ( sin(real,Xb) = zero_zero(real) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3244_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,Mb: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),aa(int,A,ring_1_of_int(A),Mb)))) ) ).

% round_diff_minimal
tff(fact_3245_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),aa(A,A,abs_abs(A),Yb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_le_square_iff
tff(fact_3246_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_3247_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),Yb) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_3248_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: fun(A,fun(A,$o)),Xb: A] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X3)
             => aa(A,$o,aa(A,fun(A,$o),P2,X3),aa(nat,A,power_power(A,X3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P2,aa(A,A,abs_abs(A),Xb)),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_3249_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_3250_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xb)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_3251_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A2),Nb)),aa(nat,A,power_power(A,B2),Nb)) ) ) ) ).

% power_mono_even
tff(fact_3252_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3253_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),Xb: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( member(A,I3,I6)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xb,I3)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Xb),I6) = one_one(B) )
           => ( ! [I3: A] :
                  ( member(A,I3,I6)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,I3)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_hy(fun(A,B),fun(fun(A,B),fun(A,B)),Xb),A2)),I6)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3254_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_3255_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_3256_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,K5: real,C3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),K5)
         => ( ! [X3: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X3)),K5)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),C3),X3)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ia(A,fun(fun(nat,A),fun(nat,A)),Xb),C3)) ) ) ) ).

% termdiff_converges
tff(fact_3257_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),pi),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_3258_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3259_sin__monotone__2pi,axiom,
    ! [Yb: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Yb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Yb)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3260_sin__mono__less__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Yb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Yb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),sin(real,Yb))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3261_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3262_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))),Xb))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_3263_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(int,A,ring_1_of_int(A),Nb)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))
         => ( archimedean_round(A,Xb) = Nb ) ) ) ).

% round_unique'
tff(fact_3264_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Yb)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_3265_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ? [T5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),Xb)
          & ( sin(real,Xb) = 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_ib(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3266_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),Xb)
            & ( sin(real,Xb) = 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_ib(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3267_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_id(A,fun(A,fun(nat,A)),Xb),Yb),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Yb))) ) ).

% sin_x_sin_y
tff(fact_3268_lemma__interval,axiom,
    ! [A2: real,Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [Y3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y3))),D2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Y3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_3269_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_ie(real,fun(nat,real),Xb)) ) ).

% monoseq_arctan_series
tff(fact_3270_arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( arctan(Xb) = suminf(real,aTP_Lamp_hr(real,fun(nat,real),Xb)) ) ) ).

% arctan_series
tff(fact_3271_lemma__interval__lt,axiom,
    ! [A2: real,Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [Y3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y3))),D2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_3272_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,B2: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb) )
        <=> ( ( A2 = B2 )
            | ( Xb = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_3273_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_3274_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),Xb) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_3275_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( Xb = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_3276_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,U: real,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,U),B2)) )
        <=> ( ( A2 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_3277_zero__less__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),arctan(Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% zero_less_arctan_iff
tff(fact_3278_arctan__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_3279_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Xb),one_one(int))
    <=> ( aa(int,int,abs_abs(int),Xb) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_3280_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),U)),A2)),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = A2 ) ).

% scaleR_collapse
tff(fact_3281_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_3282_dvd__nat__abs__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K2) ) ).

% dvd_nat_abs_iff
tff(fact_3283_nat__abs__dvd__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K2))),Nb)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K2),aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_abs_dvd_iff
tff(fact_3284_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)) = A2 ) ).

% scaleR_half_double
tff(fact_3285_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,A2: real,B2: real] :
          ( ( Xb != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_3286_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Yb: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Yb)) ) ).

% scaleR_right_distrib
tff(fact_3287_zdvd__antisym__abs,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B2),A2)
       => ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) ) ) ) ).

% zdvd_antisym_abs
tff(fact_3288_arctan__less__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xb)),arctan(Yb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ).

% arctan_less_iff
tff(fact_3289_arctan__monotone,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xb)),arctan(Yb)) ) ).

% arctan_monotone
tff(fact_3290_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ).

% scaleR_left_distrib
tff(fact_3291_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Yb: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Yb)),Xaa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Xaa)),aa(A,A,real_V8093663219630862766scaleR(A,Yb),Xaa)) ) ).

% scaleR_left.add
tff(fact_3292_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),Mb) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_3293_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S2)
    <=> ! [M2: int] :
        ? [N: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M2),aa(int,int,abs_abs(int),N))
          & member(int,N,S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_3294_infinite__int__iff__unbounded,axiom,
    ! [S2: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S2)
    <=> ! [M2: int] :
        ? [N: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M2),aa(int,int,abs_abs(int),N))
          & member(int,N,S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_3295_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C3)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C3)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3296_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scaleR_right_mono
tff(fact_3297_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_3298_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_3299_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_3300_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,Yb: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Yb)) ) ) ) ).

% scaleR_left_mono
tff(fact_3301_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C3: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B2)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_3302_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,U: real,V: real,A2: A] :
          ( ( Xb = aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V)),A2) )
        <=> $ite(V = zero_zero(real),Xb = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_3303_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V: real,A2: A,Xb: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V)),A2) = Xb )
        <=> $ite(V = zero_zero(real),Xb = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V),Xb)) ) ) ).

% vector_fraction_eq_iff
tff(fact_3304_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C3: A,B2: real,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),E3)),C3)),D3) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3305_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C3: A,B2: real,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),E3)),D3)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3306_zabs__def,axiom,
    ! [I2: int] :
      aa(int,int,abs_abs(int),I2) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),zero_zero(int)),aa(int,int,uminus_uminus(int),I2),I2) ).

% zabs_def
tff(fact_3307_dvd__imp__le__int,axiom,
    ! [I2: int,D3: int] :
      ( ( I2 != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D3),I2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D3)),aa(int,int,abs_abs(int),I2)) ) ) ).

% dvd_imp_le_int
tff(fact_3308_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z: int] : aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),W2))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_3309_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K2,L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_3310_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_3311_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_3312_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_3313_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3314_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3315_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3316_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_3317_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_3318_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C3: A,D3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C3)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D3)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3319_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A,Yb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Yb)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3320_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),Xb) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3321_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),bit0(one2))),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ).

% scaleR_2
tff(fact_3322_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K2))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_3323_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb)),Mb)
      <=> ( aa(int,int,abs_abs(int),Nb) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_3324_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) ).

% nat_abs_int_diff
tff(fact_3325_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),I3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I3))),aa(nat,int,F3,I3)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F3,Mb)),K2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(nat,int,F3,Nb))
           => ? [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),I3)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
                & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_3326_decr__lemma,axiom,
    ! [D3: int,Xb: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Z))),one_one(int))),D3))),Z) ) ).

% decr_lemma
tff(fact_3327_incr__lemma,axiom,
    ! [D3: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Z))),one_one(int))),D3))) ) ).

% incr_lemma
tff(fact_3328_arctan__ubound,axiom,
    ! [Yb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Yb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arctan_ubound
tff(fact_3329_nat__ivt__aux,axiom,
    ! [Nb: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I3))),aa(nat,int,F3,I3)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(nat,int,F3,Nb))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_3330_arctan__bounded,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Yb))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Yb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% arctan_bounded
tff(fact_3331_arctan__lbound,axiom,
    ! [Yb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Yb)) ).

% arctan_lbound
tff(fact_3332_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F3: fun(nat,int),K2: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F3,I3)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(nat,int,F3,Nb))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,int,F3,I3) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_3333_arctan__add,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Yb)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),arctan(Xb)),arctan(Yb)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Yb),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Yb)))) ) ) ) ).

% arctan_add
tff(fact_3334_divide__int__def,axiom,
    ! [K2: int,L: int] :
      divide_divide(int,K2,L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(sgn_sgn(int,K2) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K2)))))) ) ).

% divide_int_def
tff(fact_3335_arctan__double,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),arctan(Xb)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xb),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_3336_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ig(A,fun(A,fun(nat,A)),Xb),Yb),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb))) ) ).

% sums_cos_x_plus_y
tff(fact_3337_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ii(A,fun(A,fun(nat,A)),Xb),Yb),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Yb))) ) ).

% cos_x_cos_y
tff(fact_3338_sincos__total__2pi,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Yb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xb = cos(real,T5) )
               => ( Yb != sin(real,T5) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3339_sin__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( sin(real,Xb) = divide_divide(real,aa(real,real,tan(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_3340_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [M2: nat,N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M2)),aa(nat,A,X5,N)) )
            | ! [M2: nat,N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,M2)) ) ) ) ) ).

% monoseq_def
tff(fact_3341_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_3342_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_3343_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Xb))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_3344_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_3345_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_3346_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3347_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Yb) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Yb)) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Yb))) ) ) ) ) ).

% add_tan_eq
tff(fact_3348_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) = one_one(A) )
         => ( sin(A,Xb) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3349_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),cos(A,Yb))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),sin(A,Yb))) ) ).

% sin_add
tff(fact_3350_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Yb) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Yb))) = divide_divide(A,cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Yb))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3351_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Yb) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Yb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Yb)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3352_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Yb) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Yb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Yb)))) ) ) ) ) ) ).

% tan_add
tff(fact_3353_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Yb))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Yb))) ) ).

% cos_add
tff(fact_3354_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Yb))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Yb))) ) ).

% cos_diff
tff(fact_3355_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,Xb)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3356_cos__monotone__0__pi,axiom,
    ! [Yb: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Yb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xb)),cos(real,Yb)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3357_cos__mono__less__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Yb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Yb),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xb)),cos(real,Yb))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3358_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tan(A),Xb) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb))),one_one(A))) ) ).

% tan_half
tff(fact_3359_cos__two__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_3360_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,tan(A),Xb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_3361_cos__monotone__minus__pi__0,axiom,
    ! [Yb: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Yb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Yb)),cos(real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3362_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
      & ( sin(real,Y) = sin(real,Xb) )
      & ( cos(real,Y) = cos(real,Xb) ) ) ).

% sincos_principal_value
tff(fact_3363_cos__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( cos(real,Xb) = divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_3364_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,W2)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),cos(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_3365_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W2)),cos(A,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_cos
tff(fact_3366_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tan(real),Xb)) ) ) ).

% tan_gt_zero
tff(fact_3367_lemma__tan__total,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb)
     => ? [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),aa(real,real,tan(real),X3)) ) ) ).

% lemma_tan_total
tff(fact_3368_tan__total,axiom,
    ! [Yb: real] :
    ? [X3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),X3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X3) = Yb )
      & ! [Y3: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,tan(real),Y3) = Yb ) )
         => ( Y3 = X3 ) ) ) ).

% tan_total
tff(fact_3369_tan__monotone,axiom,
    ! [Yb: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Yb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Yb)),aa(real,real,tan(real),Xb)) ) ) ) ).

% tan_monotone
tff(fact_3370_tan__monotone_H,axiom,
    ! [Yb: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Yb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),Xb)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Yb)),aa(real,real,tan(real),Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3371_tan__mono__lt__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Yb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Yb))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3372_lemma__tan__total1,axiom,
    ! [Yb: real] :
    ? [X3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),X3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X3) = Yb ) ) ).

% lemma_tan_total1
tff(fact_3373_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xb))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_3374_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_3375_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),cos(A,W2)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z),aa(num,A,numeral_numeral(A),bit0(one2)))))),sin(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_3376_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),sin(A,W2)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_diff_sin
tff(fact_3377_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,W2)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_3378_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W2)),sin(A,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_sin
tff(fact_3379_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W2)),cos(A,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z))),sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_cos
tff(fact_3380_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W2: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W2)),sin(A,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),W2),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W2),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_sin
tff(fact_3381_tan__total__pos,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Yb)
     => ? [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( aa(real,real,tan(real),X3) = Yb ) ) ) ).

% tan_total_pos
tff(fact_3382_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_3383_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xb)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3384_tan__mono__le__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Yb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Yb))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Yb) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3385_tan__mono__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Yb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Yb)) ) ) ) ).

% tan_mono_le
tff(fact_3386_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),Xb))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3387_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_3388_arctan,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Yb))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Yb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),arctan(Yb)) = Yb ) ) ).

% arctan
tff(fact_3389_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( arctan(aa(real,real,tan(real),Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_3390_arctan__unique,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),Xb) = Yb )
         => ( arctan(Yb) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_3391_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,uminus_uminus(A),sin(A,Xb)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3392_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ? [Z2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z2) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_3393_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,aa(nat,nat,suc,N2)))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI1
tff(fact_3394_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N2))),aa(nat,A,X5,N2))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI2
tff(fact_3395_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,aa(nat,nat,suc,N)))
            | ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N))),aa(nat,A,X5,N)) ) ) ) ).

% monoseq_Suc
tff(fact_3396_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M)),aa(nat,A,X5,N2)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI1
tff(fact_3397_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,M)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI2
tff(fact_3398_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),Xb)
            & ( cos(real,Xb) = 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_ij(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3399_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),zero_zero(real))
            & ( cos(real,Xb) = 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_ij(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi))),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3400_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Xb)),suminf(A,aTP_Lamp_ik(A,fun(nat,A),Xb))) ) ).

% exp_first_two_terms
tff(fact_3401_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(cos(real,T5),sin(real,T5)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3402_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3) )
               => ( ( sgn_sgn(int,R3) = sgn_sgn(int,A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A22))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_3403_inverse__eq__iff__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% inverse_eq_iff_eq
tff(fact_3404_inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ).

% inverse_inverse_eq
tff(fact_3405_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_3406_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3407_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) ) ).

% inverse_mult_distrib
tff(fact_3408_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] :
          ( ( aa(A,A,inverse_inverse(A),Xb) = one_one(A) )
        <=> ( Xb = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_3409_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_3410_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),divide_divide(A,A2,B2)) = divide_divide(A,B2,A2) ) ).

% inverse_divide
tff(fact_3411_inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ).

% inverse_minus_eq
tff(fact_3412_abs__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ).

% abs_inverse
tff(fact_3413_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) ) ).

% sgn_inverse
tff(fact_3414_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% inverse_sgn
tff(fact_3415_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_3416_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_3417_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% inverse_positive_iff_positive
tff(fact_3418_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_3419_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_3420_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_3421_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3422_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_3423_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_3424_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_3425_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_3426_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_3427_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_3428_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_3429_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_3430_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_3431_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Yb: A,Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Yb)),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),Yb)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_3432_inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( A2 = B2 ) ) ) ).

% inverse_eq_imp_eq
tff(fact_3433_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_3434_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,Xb: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( Xb != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2)),aa(A,A,inverse_inverse(A),Xb)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_3435_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),real_V7770717601297561774m_norm(A,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xb))),aa(real,real,inverse_inverse(real),R2)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_3436_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ).

% positive_imp_inverse_positive
tff(fact_3437_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_3438_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_3439_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_3440_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_3441_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_3442_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less
tff(fact_3443_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less
tff(fact_3444_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_3445_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_3446_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = B2 ) ) ) ).

% inverse_unique
tff(fact_3447_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : divide_divide(A,A2,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_3448_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : divide_divide(A,A2,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_3449_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : divide_divide(A,A2,B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A2) ) ).

% divide_inverse_commute
tff(fact_3450_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),A2) = divide_divide(A,one_one(A),A2) ) ).

% inverse_eq_divide
tff(fact_3451_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Xb),Mb)),aa(A,A,inverse_inverse(A),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Xb)),aa(nat,A,power_power(A,Xb),Mb)) ) ).

% power_mult_inverse_distrib
tff(fact_3452_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Xb),Mb)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),Xb)),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),Xb)),Nb)),aa(nat,A,power_power(A,Xb),Mb)) ) ).

% power_mult_power_inverse_commute
tff(fact_3453_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: nat,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xaa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xaa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_3454_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_3455_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: int,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xaa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xaa))) ) ).

% mult_inverse_of_int_commute
tff(fact_3456_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_3457_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_3458_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le
tff(fact_3459_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le
tff(fact_3460_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% inverse_le_1_iff
tff(fact_3461_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Xb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_3462_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_less_inverse
tff(fact_3463_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_3464_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_3465_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_3466_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_3467_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_3468_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_3469_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_3470_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Xb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_3471_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ) ).

% inverse_less_1_iff
tff(fact_3472_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_le_inverse
tff(fact_3473_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_3474_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),Xb) ) ) ).

% reals_Archimedean
tff(fact_3475_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Xb: A,C3: A,Yb: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Mb),Xb)),C3) = Yb )
          <=> ( Xb = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),Yb)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),C3)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_3476_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Yb: A,Xb: A,C3: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( Yb = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Mb),Xb)),C3) )
          <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),Yb)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),C3)) = Xb ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_3477_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3478_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3479_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3480_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3481_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_3482_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_3483_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_3484_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_3485_forall__pos__mono__1,axiom,
    ! [P2: fun(real,$o),E3: real] :
      ( ! [D2: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D2),E2)
         => ( aa(real,$o,P2,D2)
           => aa(real,$o,P2,E2) ) )
     => ( ! [N2: nat] : aa(real,$o,P2,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => aa(real,$o,P2,E3) ) ) ) ).

% forall_pos_mono_1
tff(fact_3486_forall__pos__mono,axiom,
    ! [P2: fun(real,$o),E3: real] :
      ( ! [D2: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D2),E2)
         => ( aa(real,$o,P2,D2)
           => aa(real,$o,P2,E2) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => aa(real,$o,P2,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N2))) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => aa(real,$o,P2,E3) ) ) ) ).

% forall_pos_mono
tff(fact_3487_real__arch__inverse,axiom,
    ! [E3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
    <=> ? [N: nat] :
          ( ( N != zero_zero(nat) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N)))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))),E3) ) ) ).

% real_arch_inverse
tff(fact_3488_ln__inverse,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_inverse
tff(fact_3489_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N2))),Xb) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3490_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => ( aa(nat,A,power_power(A,Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Xb),Nb)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),Xb)),Mb)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3491_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3492_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3493_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3494_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3495_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_3496_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3497_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3498_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3499_log__inverse,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_3500_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,inverse_inverse(real),Xb))) ) ).

% plus_inverse_ge_2
tff(fact_3501_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(nat,real,power_power(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_3502_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Yb: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb) = aa(A,A,aa(A,fun(A,A),times_times(A),Yb),Xb) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Nb))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)),Nb)) = 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_il(A,fun(A,fun(nat,fun(nat,A))),Xb),Yb),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_3503_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_im(A,fun(nat,A),Xb))) ) ).

% exp_first_term
tff(fact_3504_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),cos(A,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_3505_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
    <=> ( ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R2) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_3506_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q2: int] :
      ( ( sgn_sgn(int,R2) = sgn_sgn(int,L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L))
       => ( ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R2) )
         => eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_3507_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,real,power_power(real,Xb),aa(int,nat,nat2,Nb)),aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Nb))))) ) ) ).

% powr_real_of_int
tff(fact_3508_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,K2: nat] : exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_in(A,fun(nat,A),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),K2))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_io(A,fun(nat,fun(nat,A)),Xb),K2))) ) ).

% exp_first_terms
tff(fact_3509_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) )
            & ( sgn_sgn(int,R5) = sgn_sgn(int,L4) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4))
            & ( 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_3510_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ip(A,fun(nat,A),Xb),sinh(A,Xb)) ) ).

% sinh_converges
tff(fact_3511_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_iq(A,fun(nat,A),Xb),cosh(A,Xb)) ) ).

% cosh_converges
tff(fact_3512_arctan__def,axiom,
    ! [Yb: real] : arctan(Yb) = the(real,aTP_Lamp_ir(real,fun(real,$o),Yb)) ).

% arctan_def
tff(fact_3513_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( sinh(real,aa(real,real,ln_ln(real),Xb)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),aa(real,real,inverse_inverse(real),Xb)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_3514_arcsin__lt__bounded,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Yb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Yb))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Yb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3515_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_3516_sinh__real__less__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),sinh(real,Yb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ).

% sinh_real_less_iff
tff(fact_3517_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_3518_sinh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_3519_sinh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sinh(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% sinh_real_pos_iff
tff(fact_3520_sinh__less__cosh__real,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),cosh(real,Xb)) ).

% sinh_less_cosh_real
tff(fact_3521_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),cosh(A,Yb))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),sinh(A,Yb))) ) ).

% cosh_add
tff(fact_3522_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),cosh(A,Yb))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),sinh(A,Yb))) ) ).

% sinh_add
tff(fact_3523_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sinh(A,Xb)),cosh(A,Xb)) = exp(A,Xb) ) ).

% sinh_plus_cosh
tff(fact_3524_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cosh(A,Xb)),sinh(A,Xb)) = exp(A,Xb) ) ).

% cosh_plus_sinh
tff(fact_3525_cosh__real__pos,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cosh(real,Xb)) ).

% cosh_real_pos
tff(fact_3526_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Yb),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Yb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_3527_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Yb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Yb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_3528_cosh__real__strict__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Yb)) ) ) ).

% cosh_real_strict_mono
tff(fact_3529_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,power_power(A,cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_3530_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_3531_arcsin__less__arcsin,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Yb),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Yb)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3532_arcsin__less__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Yb)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Yb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ) ) ).

% arcsin_less_mono
tff(fact_3533_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xb)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3534_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Yb: A] :
          ( ( cosh(A,Xb) != zero_zero(A) )
         => ( ( cosh(A,Yb) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),Xb)),aa(A,A,tanh(A),Yb)),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),Xb)),aa(A,A,tanh(A),Yb)))) ) ) ) ) ).

% tanh_add
tff(fact_3535_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sinh(A,Xb) = zero_zero(A) )
        <=> member(A,exp(A,Xb),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),one_one(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_3536_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : cosh(A,Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Z)),exp(A,aa(A,A,uminus_uminus(A),Z))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cosh_field_def
tff(fact_3537_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cosh(A,Xb) = zero_zero(A) )
        <=> ( aa(nat,A,power_power(A,exp(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_3538_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : cosh(A,Xb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% cosh_def
tff(fact_3539_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( cosh(real,aa(real,real,ln_ln(real),Xb)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,inverse_inverse(real),Xb)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_3540_old_Orec__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,fun(B,C)),Xa: product_prod(A,B)] : product_rec_prod(A,B,C,X,Xa) = the(C,product_rec_set_prod(A,B,C,X,Xa)) ).

% old.rec_prod_def
tff(fact_3541_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% xor_Suc_0_eq
tff(fact_3542_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% Suc_0_xor_eq
tff(fact_3543_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_is(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_3544_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),Bs)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(list($o),nat,size_size(list($o)),Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_3545_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_3546_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_3547_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_3548_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Xb) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_3549_The__split__eq,axiom,
    ! [A: $tType,B: $tType,Xb: A,Yb: B] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_it(A,fun(B,fun(A,fun(B,$o))),Xb),Yb))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb) ).

% The_split_eq
tff(fact_3550_the__elem__eq,axiom,
    ! [A: $tType,Xb: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = Xb ).

% the_elem_eq
tff(fact_3551_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit1(Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(Xb)) ).

% xor_nat_numerals(4)
tff(fact_3552_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xb)) ).

% xor_nat_numerals(3)
tff(fact_3553_xor__nat__numerals_I2_J,axiom,
    ! [Yb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Yb))) = aa(num,nat,numeral_numeral(nat),bit0(Yb)) ).

% xor_nat_numerals(2)
tff(fact_3554_xor__nat__numerals_I1_J,axiom,
    ! [Yb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Yb))) = aa(num,nat,numeral_numeral(nat),bit1(Yb)) ).

% xor_nat_numerals(1)
tff(fact_3555_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Yb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xb))),aa(num,A,numeral_numeral(A),bit1(Yb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Yb)))) ) ).

% xor_numerals(4)
tff(fact_3556_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Yb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit1(Xb))),aa(num,A,numeral_numeral(A),bit0(Yb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Yb)))) ) ).

% xor_numerals(6)
tff(fact_3557_floor__real__def,axiom,
    ! [Xb: real] : archim6421214686448440834_floor(real,Xb) = the(int,aTP_Lamp_iu(real,fun(int,$o),Xb)) ).

% floor_real_def
tff(fact_3558_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Mb),Nb) = $ite(
        Mb = zero_zero(nat),
        Nb,
        $ite(Nb = zero_zero(nat),Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_3559_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Mb),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb) != ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_3560_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% xor_one_eq
tff(fact_3561_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% one_xor_eq
tff(fact_3562_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_3563_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P: 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)),P) = P ).

% case_prod_Pair_iden
tff(fact_3564_floor__rat__def,axiom,
    ! [Xb: rat] : archim6421214686448440834_floor(rat,Xb) = the(int,aTP_Lamp_iv(rat,fun(int,$o),Xb)) ).

% floor_rat_def
tff(fact_3565_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I6: set(A),F3: fun(A,B),I2: A] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_iw(set(A),fun(fun(A,B),fun(A,$o)),I6),F3))),F4)
           => ( groups1027152243600224163dd_sum(A,B,F3,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)),insert(A),I2),bot_bot(set(A))))) = $ite(member(A,I2,I6),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I6)),aa(A,B,F3,I2)),groups1027152243600224163dd_sum(A,B,F3,I6)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3566_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Mb),Nb) = $ite(
        Mb = zero_zero(nat),
        Nb,
        $ite(Nb = zero_zero(nat),Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_3567_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_3568_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_3569_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_3570_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_3571_sum_Oeq__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),P: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( groups1027152243600224163dd_sum(A,B,P,I6) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),P),I6) ) ) ) ).

% sum.eq_sum
tff(fact_3572_or__nat__numerals_I2_J,axiom,
    ! [Yb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Yb))) = aa(num,nat,numeral_numeral(nat),bit1(Yb)) ).

% or_nat_numerals(2)
tff(fact_3573_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit1(Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xb)) ).

% or_nat_numerals(4)
tff(fact_3574_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),P: fun(A,B),I2: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(A,$o)),I6),P)))
         => ( groups1027152243600224163dd_sum(A,B,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),I6)) = $ite(member(A,I2,I6),groups1027152243600224163dd_sum(A,B,P,I6),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P,I2)),groups1027152243600224163dd_sum(A,B,P,I6))) ) ) ) ).

% sum.insert'
tff(fact_3575_or__nat__numerals_I1_J,axiom,
    ! [Yb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Yb))) = aa(num,nat,numeral_numeral(nat),bit1(Yb)) ).

% or_nat_numerals(1)
tff(fact_3576_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xb)) ).

% or_nat_numerals(3)
tff(fact_3577_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Yb: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit1(Xb))),aa(num,A,numeral_numeral(A),bit1(Yb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Yb)))) ) ).

% or_numerals(7)
tff(fact_3578_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Yb: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit1(Xb))),aa(num,A,numeral_numeral(A),bit0(Yb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Yb)))) ) ).

% or_numerals(6)
tff(fact_3579_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Yb: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xb))),aa(num,A,numeral_numeral(A),bit1(Yb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Yb)))) ) ).

% or_numerals(4)
tff(fact_3580_sgn__rat__def,axiom,
    ! [A2: rat] :
      sgn_sgn(rat,A2) = $ite(
        A2 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A2),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_3581_abs__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,rat,abs_abs(rat),A2) = $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),A2),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A2),A2) ).

% abs_rat_def
tff(fact_3582_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
     => ~ ! [S3: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S3)
           => ! [T5: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T5)
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S3),T5) ) ) ) ) ).

% obtain_pos_sum
tff(fact_3583_less__eq__rat__def,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xb),Yb)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% less_eq_rat_def
tff(fact_3584_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_3585_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),zero_zero(A)) = Xb ) ).

% bit.disj_zero_right
tff(fact_3586_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),I6: set(B)] : groups1027152243600224163dd_sum(B,A,G3,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ix(fun(B,A),fun(set(B),fun(B,$o)),G3),I6))) = groups1027152243600224163dd_sum(B,A,G3,I6) ) ).

% sum.non_neutral'
tff(fact_3587_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_iy(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H),I6) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G3,I6)),groups1027152243600224163dd_sum(A,B,H,I6)) ) ) ) ).

% sum.distrib_triv'
tff(fact_3588_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
               => ( aa(A,B,G3,X3) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G3,S2) = groups1027152243600224163dd_sum(A,B,G3,T2) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3589_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
               => ( aa(A,B,G3,X3) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G3,T2) = groups1027152243600224163dd_sum(A,B,G3,S2) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3590_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T2: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
         => ( ! [I3: A] :
                ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
               => ( aa(A,B,H,I3) = zero_zero(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S2)
                 => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) )
             => ( groups1027152243600224163dd_sum(A,B,G3,S2) = groups1027152243600224163dd_sum(A,B,H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3591_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T2: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T2)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
               => ( aa(A,B,G3,X3) = zero_zero(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S2)
                 => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) )
             => ( groups1027152243600224163dd_sum(A,B,G3,T2) = groups1027152243600224163dd_sum(A,B,H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3592_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Xb) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Yb) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Yb) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( Xb = Yb ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_3593_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(A,$o)),I6),G3)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(A,$o)),I6),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_iy(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H),I6) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G3,I6)),groups1027152243600224163dd_sum(A,B,H,I6)) ) ) ) ) ).

% sum.distrib'
tff(fact_3594_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P: fun(B,A),I6: set(B)] :
          groups1027152243600224163dd_sum(B,A,P,I6) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ix(fun(B,A),fun(set(B),fun(B,$o)),P),I6))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ix(fun(B,A),fun(set(B),fun(B,$o)),P),I6))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_3595_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F3: fun(fun(A,B),C),G3: C] :
          ( ! [X3: fun(A,B)] : aa(fun(A,B),C,F3,X3) = G3
         => ( aa(fun(A,B),C,F3,aTP_Lamp_iz(A,B)) = G3 ) ) ) ).

% fun_cong_unused_0
tff(fact_3596_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl_dec(A,Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_3597_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% one_or_eq
tff(fact_3598_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% or_one_eq
tff(fact_3599_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Yb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Yb),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xb),Yb)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_3600_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I6: set(A),F3: fun(A,B),I2: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_iw(set(A),fun(fun(A,B),fun(A,$o)),I6),F3)))
         => ( groups1027152243600224163dd_sum(A,B,F3,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)),insert(A),I2),bot_bot(set(A))))) = $ite(member(A,I2,I6),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I6)),aa(A,B,F3,I2)),groups1027152243600224163dd_sum(A,B,F3,I6)) ) ) ) ).

% sum_diff1'
tff(fact_3601_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% or_Suc_0_eq
tff(fact_3602_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% Suc_0_or_eq
tff(fact_3603_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Mb),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb)
            | ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_3604_normalize__negative,axiom,
    ! [Q2: int,P: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q2),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_3605_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ca(nat,nat)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_3606_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_3607_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
         => ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,X5,M2)),aa(nat,real,X5,N)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_3608_sum__power2,axiom,
    ! [K2: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)),one_one(nat)) ).

% sum_power2
tff(fact_3609_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_3610_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or7035219750837199246ssThan(nat,L,U)) ).

% finite_atLeastLessThan
tff(fact_3611_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( member(A,I2,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_3612_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_3613_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,J: A,Mb: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I2,J)),set_or7035219750837199246ssThan(A,Mb,Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),I2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Nb) ) ) ) ) ).

% ivl_subset
tff(fact_3614_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_3615_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_3616_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_3617_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,Nb: A,Mb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),Nb)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I2,Mb)),set_or7035219750837199246ssThan(A,I2,Nb)) = set_or7035219750837199246ssThan(A,Nb,Mb) ) ) ) ).

% ivl_diff
tff(fact_3618_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Nb: A,Mb: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(A,set(A),set_ord_lessThan(A),Nb)),aa(A,set(A),set_ord_lessThan(A),Mb)) = set_or7035219750837199246ssThan(A,Mb,Nb) ) ).

% lessThan_minus_lessThan
tff(fact_3619_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Mb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Mb),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_3620_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G3,Nb))) ) ).

% sum.op_ivl_Suc
tff(fact_3621_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G3,Nb))) ) ).

% prod.op_ivl_Suc
tff(fact_3622_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
            <=> ( ( A2 = C3 )
                & ( B2 = D3 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_3623_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
             => ( A2 = C3 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_3624_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
             => ( B2 = D3 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_3625_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_3626_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_3627_all__nat__less__eq,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
         => aa(nat,$o,P2,M2) )
    <=> ! [X4: nat] :
          ( member(nat,X4,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P2,X4) ) ) ).

% all_nat_less_eq
tff(fact_3628_ex__nat__less__eq,axiom,
    ! [Nb: nat,P2: fun(nat,$o)] :
      ( ? [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
          & aa(nat,$o,P2,M2) )
    <=> ? [X4: nat] :
          ( member(nat,X4,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P2,X4) ) ) ).

% ex_nat_less_eq
tff(fact_3629_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_3630_lessThan__atLeast0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_lessThan(nat),Nb) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ).

% lessThan_atLeast0
tff(fact_3631_atLeastLessThan0,axiom,
    ! [Mb: nat] : set_or7035219750837199246ssThan(nat,Mb,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_3632_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ci(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_3633_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_cj(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_3634_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ec(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_3635_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ed(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_3636_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C3: A,B2: A,D3: A,G3: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C3 )
         => ( ( B2 = D3 )
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D3)
                   => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),set_or7035219750837199246ssThan(A,C3,D3)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_3637_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C3: A,B2: A,D3: A,G3: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C3 )
         => ( ( B2 = D3 )
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D3)
                   => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),set_or7035219750837199246ssThan(A,C3,D3)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_3638_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,P: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P)
           => ( 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),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Nb,P))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,P)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_3639_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,P: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P)
           => ( 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),F3),set_or7035219750837199246ssThan(nat,Mb,P))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,Mb,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,Nb,P)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_3640_size__list__estimation,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: nat,F3: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Yb),aa(A,nat,F3,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Yb),aa(list(A),nat,size_list(A,F3),Xs)) ) ) ).

% size_list_estimation
tff(fact_3641_size__list__estimation_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: nat,F3: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Yb),aa(A,nat,F3,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Yb),aa(list(A),nat,size_list(A,F3),Xs)) ) ) ).

% size_list_estimation'
tff(fact_3642_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat),G3: fun(A,nat)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,X3)),aa(A,nat,G3,X3)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_list(A,F3),Xs)),aa(list(A),nat,size_list(A,G3),Xs)) ) ).

% size_list_pointwise
tff(fact_3643_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,P: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P)
           => ( 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),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Nb,P))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,P)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_3644_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_3645_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(set(nat),$o,finite_finite2(nat),N3) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_3646_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3647_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3648_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_3649_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G3,Nb)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_3650_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_3651_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),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),G3),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_3652_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)),insert(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3653_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G3,Nb)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_3654_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_3655_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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),G3),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_3656_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% sum.last_plus
tff(fact_3657_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% prod.last_plus
tff(fact_3658_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_co(fun(nat,A),fun(nat,A),F3)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,Mb)) ) ) ) ).

% sum_Suc_diff'
tff(fact_3659_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ja(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% sum.atLeastLessThan_rev
tff(fact_3660_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Nb),set_or7035219750837199246ssThan(nat,Mb,Nb)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_3661_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jb(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fk(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% sum.nested_swap
tff(fact_3662_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% prod.atLeastLessThan_rev
tff(fact_3663_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jd(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% prod.nested_swap
tff(fact_3664_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_je(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))) ) ).

% sum.nat_group
tff(fact_3665_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))) ) ).

% prod.nat_group
tff(fact_3666_prod__Suc__Suc__fact,axiom,
    ! [Nb: 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)),Nb)) = semiring_char_0_fact(nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_3667_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) = semiring_char_0_fact(nat,Nb) ).

% prod_Suc_fact
tff(fact_3668_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G3,Nb))) ) ).

% sum.head_if
tff(fact_3669_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G3,Nb))) ) ).

% prod.head_if
tff(fact_3670_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P: int,Q2: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% normalize_denom_pos
tff(fact_3671_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% fact_prod_Suc
tff(fact_3672_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_cl(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3673_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_eg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3674_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ei(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_prod
tff(fact_3675_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K2: num] :
      set_or7035219750837199246ssThan(nat,Mb,aa(num,nat,numeral_numeral(nat),K2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),pred_numeral(K2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),pred_numeral(K2)),set_or7035219750837199246ssThan(nat,Mb,pred_numeral(K2))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_3676_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aa(nat,fun(nat,nat),minus_minus(nat),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% fact_prod_rev
tff(fact_3677_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [N6: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M2)
                 => ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M2,N)))),E4) ) ) ) ) ).

% summable_Cauchy
tff(fact_3678_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ? [M7: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M3)
               => ! [N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N4)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,M3)),aa(nat,A,X5,N4)))),E3) ) ) ) ) ) ).

% CauchyD
tff(fact_3679_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,M)),aa(nat,A,X5,N2)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_3680_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M8: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,M2)),aa(nat,A,X5,N)))),E4) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_3681_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A),S: A,K2: nat] :
          ( sums(A,F3,S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,A),fun(nat,fun(nat,A)),F3),K2),S) ) ) ) ).

% sums_group
tff(fact_3682_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Yb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Yb),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Yb)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ) ).

% OR_upper
tff(fact_3683_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3684_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2),Nb)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2))) ) ) ) ).

% fact_split
tff(fact_3685_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jh(nat,fun(nat,fun(nat,A)),K2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_3686_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),K2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ji(A,fun(nat,fun(nat,A)),A2),K2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_altdef_of_nat
tff(fact_3687_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K2)),semiring_char_0_fact(A,K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jj(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_mult_fact'
tff(fact_3688_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),aa(nat,A,gbinomial(A,A2),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jj(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_mult_fact
tff(fact_3689_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),K2) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hq(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_prod_rev
tff(fact_3690_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,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_jk(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3691_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I3: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I3)),aa(nat,A,A2,J2)) ) )
         => ( ! [I3: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I3)) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_jl(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))),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),Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_3692_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I3: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A2,I3)),aa(nat,nat,A2,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I3)) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_jm(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))),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),Nb))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_3693_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_3694_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),$o,finite_finite2(int),set_or7035219750837199246ssThan(int,L,U)) ).

% finite_atLeastLessThan_int
tff(fact_3695_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),$o,finite_finite2(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_3696_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_3697_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_3698_is__singleton__the__elem,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
    <=> ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the_elem(A,A5)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_3699_length__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),subseqs(A,Xs)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_subseqs
tff(fact_3700_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_jn(set(A),fun(fun(A,B),fun(A,$o)),A5),F3)))) ) ) ) ).

% even_sum_iff
tff(fact_3701_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),Bs)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),Nb) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3702_card__lessThan,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_lessThan(nat),U)) = U ).

% card_lessThan
tff(fact_3703_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_3704_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_3705_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_atMost(nat),U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_3706_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_atLeastLessThan
tff(fact_3707_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Nb))) = aa(nat,nat,suc,Nb) ).

% card_Collect_le_nat
tff(fact_3708_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_3709_card_Oinfinite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_3710_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,U)),L) ).

% card_atLeastAtMost
tff(fact_3711_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_3712_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_atLeastLessThan_int
tff(fact_3713_is__singletonI,axiom,
    ! [A: $tType,Xb: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_3714_card__0__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) )
      <=> ( A5 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_3715_card__insert__disjoint,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ~ member(A,Xb,A5)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A5)) ) ) ) ).

% card_insert_disjoint
tff(fact_3716_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] :
      ( member(A,A2,A5)
     => ( ~ member(A,A2,B5)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_3717_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or1337092689740270186AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_3718_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% bit_0
tff(fact_3719_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)
        <=> ( ( Nb = zero_zero(nat) )
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_3720_subseqs__refl,axiom,
    ! [A: $tType,Xs: list(A)] : member(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) ).

% subseqs_refl
tff(fact_3721_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [N2: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N2)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N2) )
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
              | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_3722_is__singleton__altdef,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
    <=> ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_3723_n__subsets,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(nat,fun(set(A),$o),aTP_Lamp_jo(set(A),fun(nat,fun(set(A),$o)),A5),K2))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A5)),K2) ) ) ).

% n_subsets
tff(fact_3724_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),Nb)
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_3725_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_3726_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_3727_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N2: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N2)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ) ) ).

% disjunctive_add
tff(fact_3728_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A2)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_take_bit_iff
tff(fact_3729_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: $o,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa($o,A,zero_neq_one_of_bool(A),(B2))),Nb)
        <=> ( (B2)
            & ( Nb = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_3730_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ? [B8: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B8)
          & ( aa(set(A),nat,finite_card(A),B8) = Nb )
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B8),A5) ) ) ).

% infinite_arbitrarily_large
tff(fact_3731_card__subset__eq,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( ( aa(set(A),nat,finite_card(A),A5) = aa(set(A),nat,finite_card(A),B5) )
         => ( A5 = B5 ) ) ) ) ).

% card_subset_eq
tff(fact_3732_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B5: set(A),A5: set(B),R2: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( ! [A4: B] :
            ( member(B,A4,A5)
           => ? [B9: A] :
                ( member(A,B9,B5)
                & aa(A,$o,aa(B,fun(A,$o),R2,A4),B9) ) )
       => ( ! [A12: B,A23: B,B4: A] :
              ( member(B,A12,A5)
             => ( member(B,A23,A5)
               => ( member(A,B4,B5)
                 => ( aa(A,$o,aa(B,fun(A,$o),R2,A12),B4)
                   => ( aa(A,$o,aa(B,fun(A,$o),R2,A23),B4)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3733_card__insert__le,axiom,
    ! [A: $tType,A5: set(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5))) ).

% card_insert_le
tff(fact_3734_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) = aa(int,nat,nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_3735_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S: set(A),Ta: set(B),R: fun(A,fun(B,$o)),K2: fun(B,nat)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(B),$o,finite_finite2(B),Ta)
       => ( ! [X3: B] :
              ( member(B,X3,Ta)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_am(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S),R),X3))) = aa(B,nat,K2,X3) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_jp(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Ta),R)),S) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K2),Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_3736_card__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_au(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) = aa(nat,nat,power_power(nat,aa(set(A),nat,finite_card(A),A5)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_3737_is__singletonI_H,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ! [X3: A,Y: A] :
            ( member(A,X3,A5)
           => ( member(A,Y,A5)
             => ( X3 = Y ) ) )
       => is_singleton(A,A5) ) ) ).

% is_singletonI'
tff(fact_3738_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_3739_card__2__iff_H,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X4: A] :
          ( member(A,X4,S2)
          & ? [Xa2: A] :
              ( member(A,Xa2,S2)
              & ( X4 != Xa2 )
              & ! [Xb4: A] :
                  ( member(A,Xb4,S2)
                 => ( ( Xb4 = X4 )
                    | ( Xb4 = Xa2 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_3740_card__eq__0__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) )
    <=> ( ( A5 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% card_eq_0_iff
tff(fact_3741_card__ge__0__finite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
     => aa(set(A),$o,finite_finite2(A),A5) ) ).

% card_ge_0_finite
tff(fact_3742_card__Suc__eq__finite,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K2) )
    <=> ? [B7: A,B10: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B7),B10) )
          & ~ member(A,B7,B10)
          & ( aa(set(A),nat,finite_card(A),B10) = K2 )
          & aa(set(A),$o,finite_finite2(A),B10) ) ) ).

% card_Suc_eq_finite
tff(fact_3743_card__insert__if,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = $ite(member(A,Xb,A5),aa(set(A),nat,finite_card(A),A5),aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A5))) ) ) ).

% card_insert_if
tff(fact_3744_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C5: nat] :
      ( ! [G4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G4),F4)
         => ( aa(set(A),$o,finite_finite2(A),G4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G4)),C5) ) )
     => ( aa(set(A),$o,finite_finite2(A),F4)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F4)),C5) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_3745_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Nb: nat,S2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),S2))
     => ~ ! [T4: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T4),S2)
           => ( ( aa(set(A),nat,finite_card(A),T4) = Nb )
             => ~ aa(set(A),$o,finite_finite2(A),T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3746_card__seteq,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B5)),aa(set(A),nat,finite_card(A),A5))
         => ( A5 = B5 ) ) ) ) ).

% card_seteq
tff(fact_3747_card__mono,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ).

% card_mono
tff(fact_3748_card__less__sym__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3749_card__le__sym__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3750_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% card_length
tff(fact_3751_card__1__singletonE,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) )
     => ~ ! [X3: A] : A5 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_3752_psubset__card__mono,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ).

% psubset_card_mono
tff(fact_3753_card__less__Suc2,axiom,
    ! [M4: set(nat),I2: nat] :
      ( ~ member(nat,zero_zero(nat),M4)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jq(set(nat),fun(nat,fun(nat,$o)),M4),I2))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jr(set(nat),fun(nat,fun(nat,$o)),M4),I2))) ) ) ).

% card_less_Suc2
tff(fact_3754_card__less__Suc,axiom,
    ! [M4: set(nat),I2: nat] :
      ( member(nat,zero_zero(nat),M4)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jq(set(nat),fun(nat,fun(nat,$o)),M4),I2)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jr(set(nat),fun(nat,fun(nat,$o)),M4),I2))) ) ) ).

% card_less_Suc
tff(fact_3755_card__less,axiom,
    ! [M4: set(nat),I2: nat] :
      ( member(nat,zero_zero(nat),M4)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_jr(set(nat),fun(nat,fun(nat,$o)),M4),I2))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3756_sum__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_js(fun(A,nat),fun(A,nat),F3)),A5) = 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),F3),A5)),aa(set(A),nat,finite_card(A),A5)) ).

% sum_Suc
tff(fact_3757_subset__card__intvl__is__intvl,axiom,
    ! [A5: set(nat),K2: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A5),set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(set(nat),nat,finite_card(nat),A5))))
     => ( A5 = set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(set(nat),nat,finite_card(nat),A5))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_3758_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T2: set(B),R: fun(A,fun(B,$o)),K2: nat] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),T2)
       => ( ! [X3: B] :
              ( member(B,X3,T2)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_am(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S2),R),X3))) = K2 ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_jp(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T2),R)),S2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(set(B),nat,finite_card(B),T2)) ) ) ) ) ).

% sum_multicount
tff(fact_3759_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),F3: fun(A,B),K5: B] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),K5) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5))),K5)) ) ) ).

% sum_bounded_above
tff(fact_3760_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),K5: B,F3: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K5),aa(A,B,F3,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5))),K5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)) ) ) ).

% sum_bounded_below
tff(fact_3761_card__gt__0__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
    <=> ( ( A5 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% card_gt_0_iff
tff(fact_3762_card__1__singleton__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X4: A] : A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_3763_card__eq__SucD,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K2) )
     => ? [B4: A,B8: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),B8) )
          & ~ member(A,B4,B8)
          & ( aa(set(A),nat,finite_card(A),B8) = K2 )
          & ( ( K2 = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_3764_card__Suc__eq,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K2) )
    <=> ? [B7: A,B10: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B7),B10) )
          & ~ member(A,B7,B10)
          & ( aa(set(A),nat,finite_card(A),B10) = K2 )
          & ( ( K2 = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_3765_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X4: A] :
            ( member(A,X4,A5)
           => ! [Xa2: A] :
                ( member(A,Xa2,A5)
               => ( X4 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3766_card__le__Suc__iff,axiom,
    ! [A: $tType,Nb: nat,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),A5))
    <=> ? [A7: A,B10: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A7),B10) )
          & ~ member(A,A7,B10)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),B10))
          & aa(set(A),$o,finite_finite2(A),B10) ) ) ).

% card_le_Suc_iff
tff(fact_3767_card__Diff1__le,axiom,
    ! [A: $tType,A5: set(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)) ).

% card_Diff1_le
tff(fact_3768_card__Diff__subset,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ).

% card_Diff_subset
tff(fact_3769_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),K2)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3770_card__psubset,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5) ) ) ) ).

% card_psubset
tff(fact_3771_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ) ).

% diff_card_le_card_Diff
tff(fact_3772_card__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_av(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(set(A),nat,finite_card(A),A5))),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ) ).

% card_lists_length_le
tff(fact_3773_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ax(nat,fun(A,$o),Nb)))),Nb) ) ) ).

% card_roots_unity
tff(fact_3774_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N3)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_3775_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_3776_card__sum__le__nat__sum,axiom,
    ! [S2: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ca(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_ca(nat,nat)),S2)) ).

% card_sum_le_nat_sum
tff(fact_3777_card__nth__roots,axiom,
    ! [C3: complex,Nb: nat] :
      ( ( C3 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_jt(complex,fun(nat,fun(complex,$o)),C3),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_3778_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_cn(nat,fun(complex,$o),Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_3779_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M3)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),M3)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),N2) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),N2) ) ) ) ).

% int_bit_bound
tff(fact_3780_is__singleton__def,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
    <=> ? [X4: A] : A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_3781_is__singletonE,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
     => ~ ! [X3: A] : A5 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_3782_card__2__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X4: A,Y4: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y4),bot_bot(set(A)))) )
          & ( X4 != Y4 ) ) ) ).

% card_2_iff
tff(fact_3783_card__3__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
    <=> ? [X4: A,Y4: A,Z5: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z5),bot_bot(set(A))))) )
          & ( X4 != Y4 )
          & ( Y4 != Z5 )
          & ( X4 != Z5 ) ) ) ).

% card_3_iff
tff(fact_3784_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),A5))
     => ( A5 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_3785_card_Oremove,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( member(A,Xb,A5)
       => ( aa(set(A),nat,finite_card(A),A5) = 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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_3786_card_Oinsert__remove,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = 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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_3787_card__Suc__Diff1,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( member(A,Xb,A5)
       => ( 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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_Suc_Diff1
tff(fact_3788_card__Diff1__less,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( member(A,Xb,A5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)) ) ) ).

% card_Diff1_less
tff(fact_3789_card__Diff2__less,axiom,
    ! [A: $tType,A5: set(A),Xb: A,Yb: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( member(A,Xb,A5)
       => ( member(A,Yb,A5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)) ) ) ) ).

% card_Diff2_less
tff(fact_3790_card__Diff1__less__iff,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))
    <=> ( aa(set(A),$o,finite_finite2(A),A5)
        & member(A,Xb,A5) ) ) ).

% card_Diff1_less_iff
tff(fact_3791_card__Diff__singleton__if,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))) = $ite(member(A,Xb,A5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)),aa(set(A),nat,finite_card(A),A5)) ).

% card_Diff_singleton_if
tff(fact_3792_card__Diff__singleton,axiom,
    ! [A: $tType,Xb: A,A5: set(A)] :
      ( member(A,Xb,A5)
     => ( aa(set(A),nat,finite_card(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)),insert(A),Xb),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_3793_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = zero_zero(A) )
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_3794_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),Nb: B,K2: nat] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I3))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),Nb) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),K2)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),Nb)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(nat,B,power_power(B,Nb),K2)) ) ) ) ) ).

% prod_le_power
tff(fact_3795_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),F3: fun(A,B),K5: B] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,I3)),K5) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5))),K5)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3796_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A5: set(A),F3: fun(A,B),K5: B] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),divide_divide(B,K5,aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5)))) )
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( ( A5 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),K5) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_3797_card__insert__le__m1,axiom,
    ! [A: $tType,Nb: nat,Yb: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Yb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),Yb))),Nb) ) ) ).

% card_insert_le_m1
tff(fact_3798_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,Nb: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_fy(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb)))),Nb) ) ) ) ).

% polyfun_roots_card
tff(fact_3799_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_ju(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C3)),S2) = $ite(member(A,A2,S2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(nat,B,power_power(B,C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),S2)),one_one(nat)))),aa(nat,B,power_power(B,C3),aa(set(A),nat,finite_card(A),S2))) ) ) ) ).

% prod_gen_delta
tff(fact_3800_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3801_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_3802_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,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),bit0(one2))),B2))),Nb)
          <=> $ite(Nb = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),Nb)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3803_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,Nb: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_fy(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_fy(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb)))),Nb) ) ) ) ) ).

% polyfun_rootbound
tff(fact_3804_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> $ite(Nb = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,bit_se5641148757651400278ts_bit(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_3805_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(set(A),nat,finite_card(A),A5))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_jv(set(A),fun(nat,fun(list(A),$o)),A5),K2))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ca(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),A5)),K2)),one_one(nat)),aa(set(A),nat,finite_card(A),A5))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_3806_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ! [X3: list(A)] :
          ( member(list(A),X3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X3) = Nb ) )
     => ( aa(list(A),nat,size_size(list(A)),concat(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),Nb) ) ) ).

% length_mul_elem
tff(fact_3807_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K2: nat,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(set(A),nat,finite_card(A),A5))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_jw(nat,fun(set(A),fun(list(A),$o)),K2),A5))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ca(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),A5)),K2)),one_one(nat)),aa(set(A),nat,finite_card(A),A5))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_3808_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,A),V: num,Nb: nat] : case_nat(A,A2,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),Nb)) ).

% case_nat_add_eq_if
tff(fact_3809_distinct__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( distinct(A,list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I2)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_3810_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_jv(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_3811_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(nat,B),Nat: nat] : aa(B,A,H,case_nat(B,F1,F22,Nat)) = case_nat(A,aa(B,A,H,F1),aa(fun(nat,B),fun(nat,A),aTP_Lamp_jx(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_3812_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_3813_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_3814_subseqs__distinctD,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))
     => ( distinct(A,Xs)
       => distinct(A,Ys) ) ) ).

% subseqs_distinctD
tff(fact_3815_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X2: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X2)) = aa(nat,A,F22,X2) ).

% old.nat.simps(5)
tff(fact_3816_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_3817_finite__distinct__list,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A5 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_3818_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_jy(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_3819_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_jz(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_3820_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xs),I4) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_3821_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I2 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_3822_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_3823_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_3824_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_3825_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),aa(nat,nat,suc,Nb)) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_ka(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc2
tff(fact_3826_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),Mb) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_kb(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc1
tff(fact_3827_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ? [X3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X3) = Xb )
            & ! [Y3: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y3),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y3) = Xb ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_3828_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_ca(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ).

% diff_Suc
tff(fact_3829_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xb: A,F3: fun(nat,A),Nb: nat] :
      case_nat(A,Xb,F3,Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_3830_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I2: nat] :
      ( distinct(A,Xs)
     => ( ~ 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)),insert(A),aa(nat,A,nth(A,Xs),I2)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,I2,A2)) ) ) ).

% distinct_list_update
tff(fact_3831_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xb)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xs),Nb)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_3832_set__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),$o),aTP_Lamp_kc(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ).

% set_n_lists
tff(fact_3833_distinct__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,union(A,Xs,Ys))
    <=> distinct(A,Ys) ) ).

% distinct_union
tff(fact_3834_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P2: fun(A,$o),F1: A,F22: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P2,case_nat(A,F1,F22,Nat))
    <=> ( ( ( Nat = zero_zero(nat) )
         => aa(A,$o,P2,F1) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => aa(A,$o,P2,aa(nat,A,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_3835_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P2: fun(A,$o),F1: A,F22: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P2,case_nat(A,F1,F22,Nat))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ aa(A,$o,P2,F1) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ aa(A,$o,P2,aa(nat,A,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_3836_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_ca(nat,nat),Nat) ).

% pred_def
tff(fact_3837_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),Nb: nat,Xs: list(A)] :
      ( member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)))
     => ( aa(list(A),nat,size_size(list(A)),Ys) = Nb ) ) ).

% length_n_lists_elem
tff(fact_3838_card__Pow,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow2(A,A5)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),A5)) ) ) ).

% card_Pow
tff(fact_3839_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,fun(A,A)),V: num,Nb: nat] :
      aa(nat,A,rec_nat(A,A2,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = $let(
        pv: nat,
        pv:= pred_numeral(V),
        aa(A,A,aa(nat,fun(A,A),F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb)),aa(nat,A,rec_nat(A,A2,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb))) ) ).

% rec_nat_add_eq_if
tff(fact_3840_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = $let(
            l: A,
            l:= aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A2),
            $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,l),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_3841_wmin__insertI,axiom,
    ! [Xb: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Yb: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( member(product_prod(nat,nat),Xb,XS)
     => ( 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)),Xb),Yb),fun_pair_leq)
       => ( 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)
         => 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))),insert(product_prod(nat,nat)),Yb),YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_3842_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_3843_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_3844_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_3845_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),A2) ) ).

% push_bit_push_bit
tff(fact_3846_Pow__empty,axiom,
    ! [A: $tType] : pow2(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_3847_Pow__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),Y5: set(A)] :
      ( ( pow2(A,X5) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Y5),bot_bot(set(set(A)))) )
    <=> ( ( X5 = bot_bot(set(A)) )
        & ( Y5 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_3848_old_Onat_Osimps_I7_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,fun(A,A)),Nat: nat] : aa(nat,A,rec_nat(A,F1,F22),aa(nat,nat,suc,Nat)) = aa(A,A,aa(nat,fun(A,A),F22,Nat),aa(nat,A,rec_nat(A,F1,F22),Nat)) ).

% old.nat.simps(7)
tff(fact_3849_old_Onat_Osimps_I6_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,fun(A,A))] : aa(nat,A,rec_nat(A,F1,F22),zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_3850_finite__Pow__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),pow2(A,A5))
    <=> aa(set(A),$o,finite_finite2(A),A5) ) ).

% finite_Pow_iff
tff(fact_3851_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ).

% push_bit_of_Suc_0
tff(fact_3852_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2))
        <=> ( ( Nb != zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_3853_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_add
tff(fact_3854_Pow__bottom,axiom,
    ! [A: $tType,B5: set(A)] : member(set(A),bot_bot(set(A)),pow2(A,B5)) ).

% Pow_bottom
tff(fact_3855_Pow__not__empty,axiom,
    ! [A: $tType,A5: set(A)] : pow2(A,A5) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_3856_wmin__emptyI,axiom,
    ! [X5: set(product_prod(nat,nat))] : 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))),X5),bot_bot(set(product_prod(nat,nat)))),fun_min_weak) ).

% wmin_emptyI
tff(fact_3857_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),A2)) ) ).

% push_bit_take_bit
tff(fact_3858_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K2: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,Mb),K2)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_3859_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Mb),Q2)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3860_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_3861_binomial__def,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,nat,binomial(Nb),K2) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),$o),aTP_Lamp_kd(nat,fun(nat,fun(set(nat),$o)),Nb),K2))) ).

% binomial_def
tff(fact_3862_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ke(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% take_bit_sum
tff(fact_3863_old_Orec__nat__def,axiom,
    ! [A: $tType,X: A,Xa: fun(nat,fun(A,A)),Xb2: nat] : aa(nat,A,rec_nat(A,X,Xa),Xb2) = the(A,rec_set_nat(A,X,Xa,Xb2)) ).

% old.rec_nat_def
tff(fact_3864_wmax__insertI,axiom,
    ! [Yb: product_prod(nat,nat),YS: set(product_prod(nat,nat)),Xb: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
      ( member(product_prod(nat,nat),Yb,YS)
     => ( 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)),Xb),Yb),fun_pair_leq)
       => ( 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)
         => 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))),insert(product_prod(nat,nat)),Xb),XS)),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_3865_smax__insertI,axiom,
    ! [Yb: product_prod(nat,nat),Y5: set(product_prod(nat,nat)),Xb: product_prod(nat,nat),X5: set(product_prod(nat,nat))] :
      ( member(product_prod(nat,nat),Yb,Y5)
     => ( 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)),Xb),Yb),fun_pair_less)
       => ( 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))),X5),Y5),fun_max_strict)
         => 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))),insert(product_prod(nat,nat)),Xb),X5)),Y5),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_3866_smin__insertI,axiom,
    ! [Xb: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Yb: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
      ( member(product_prod(nat,nat),Xb,XS)
     => ( 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)),Xb),Yb),fun_pair_less)
       => ( 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)
         => 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))),insert(product_prod(nat,nat)),Yb),YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_3867_wmax__emptyI,axiom,
    ! [X5: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,finite_finite2(product_prod(nat,nat)),X5)
     => 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)))),X5),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_3868_smin__emptyI,axiom,
    ! [X5: set(product_prod(nat,nat))] :
      ( ( X5 != bot_bot(set(product_prod(nat,nat))) )
     => 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))),X5),bot_bot(set(product_prod(nat,nat)))),fun_min_strict) ) ).

% smin_emptyI
tff(fact_3869_smax__emptyI,axiom,
    ! [Y5: set(product_prod(nat,nat))] :
      ( aa(set(product_prod(nat,nat)),$o,finite_finite2(product_prod(nat,nat)),Y5)
     => ( ( Y5 != bot_bot(set(product_prod(nat,nat))) )
       => 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_3870_rec__nat__0__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F3 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F3,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_3871_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,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_3872_prod__decode__aux_Osimps,axiom,
    ! [K2: nat,Mb: nat] :
      nat_prod_decode_aux(K2,Mb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),K2),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),Mb)),nat_prod_decode_aux(aa(nat,nat,suc,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,K2)))) ).

% prod_decode_aux.simps
tff(fact_3873_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Yb )
     => ( Yb = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xb),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_3874_Suc__0__div__numeral,axiom,
    ! [K2: num] : divide_divide(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K2)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K2)) ).

% Suc_0_div_numeral
tff(fact_3875_Suc__0__mod__numeral,axiom,
    ! [K2: num] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K2)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K2)) ).

% Suc_0_mod_numeral
tff(fact_3876_finite__enumerate,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => ? [R3: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R3,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(nat),nat,finite_card(nat),S2)))
          & ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(set(nat),nat,finite_card(nat),S2))
             => member(nat,aa(nat,nat,R3,N4),S2) ) ) ) ).

% finite_enumerate
tff(fact_3877_nth__rotate1,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate1(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,Nb),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_3878_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_3879_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_3880_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_3881_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_3882_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( aa(list(A),list(A),rotate1(A),Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_3883_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o)),Xb: A,Yb: B,A2: product_prod(A,B)] :
      ( aa(B,$o,aa(A,fun(B,$o),P2,Xb),Yb)
     => ( ( A2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb) )
       => aa(B,$o,aa(A,fun(B,$o),P2,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_3884_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,Ta: product_prod(A,B)] : Ta = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Ta)),aa(product_prod(A,B),B,product_snd(A,B),Ta)) ).

% surjective_pairing
tff(fact_3885_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_3886_fst__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Yb: 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),Xb),Yb)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_3887_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X2: 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),X2)) = X1 ).

% fst_conv
tff(fact_3888_snd__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: B,Yb: 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),Xb),Yb)) = A2 )
     => ( Yb = A2 ) ) ).

% snd_eqD
tff(fact_3889_snd__conv,axiom,
    ! [B: $tType,A: $tType,X1: B,X2: A] : aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),X2)) = X2 ).

% snd_conv
tff(fact_3890_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,C: $tType,B: $tType,P2: fun(A,$o),F3: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P2,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),Prod))
    <=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
          & ~ aa(A,$o,P2,aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_3891_prod_Osplit__sel,axiom,
    ! [A: $tType,C: $tType,B: $tType,P2: fun(A,$o),F3: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P2,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),Prod))
    <=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
       => aa(A,$o,P2,aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel
tff(fact_3892_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),A5: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),Xb,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A5)))
     => aa(B,$o,aa(A,fun(B,$o),A5,aa(product_prod(A,B),A,product_fst(A,B),Xb)),aa(product_prod(A,B),B,product_snd(A,B),Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_3893_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),P: product_prod(B,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),F3),P) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),P)),aa(product_prod(B,C),C,product_snd(B,C),P)) ).

% case_prod_beta
tff(fact_3894_split__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),Prod: product_prod(B,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),F3),Prod) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) ).

% split_beta
tff(fact_3895_prod_Oexpand,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
      ( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_3896_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B),Q2: product_prod(A,B)] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),P) = aa(product_prod(A,B),A,product_fst(A,B),Q2) )
     => ( ( aa(product_prod(A,B),B,product_snd(A,B),P) = aa(product_prod(A,B),B,product_snd(A,B),Q2) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
tff(fact_3897_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,S: product_prod(A,B),Ta: product_prod(A,B)] :
      ( ( S = Ta )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S) = aa(product_prod(A,B),A,product_fst(A,B),Ta) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),S) = aa(product_prod(A,B),B,product_snd(A,B),Ta) ) ) ) ).

% prod_eq_iff
tff(fact_3898_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,fun(B,C)),Xa: product_prod(A,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),X),Xa) = aa(B,C,aa(A,fun(B,C),X,aa(product_prod(A,B),A,product_fst(A,B),Xa)),aa(product_prod(A,B),B,product_snd(A,B),Xa)) ).

% case_prod_unfold
tff(fact_3899_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),X: product_prod(A,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),F3),X) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),X)),aa(product_prod(A,B),B,product_snd(A,B),X)) ).

% case_prod_beta'
tff(fact_3900_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,F3: fun(D,fun(B,C)),G3: fun(A,D)] : aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_kf(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),F3),G3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_kg(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),F3),G3)) ).

% split_comp_eq
tff(fact_3901_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o))] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P2)) = the(product_prod(A,B),aTP_Lamp_kh(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P2)) ).

% The_case_prod
tff(fact_3902_snd__def,axiom,
    ! [A: $tType,B: $tType,Prod: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Prod) = aa(product_prod(B,A),A,aa(fun(B,fun(A,A)),fun(product_prod(B,A),A),product_case_prod(B,A,A),aTP_Lamp_ki(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_3903_fst__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,aa(fun(A,fun(B,A)),fun(product_prod(A,B),A),product_case_prod(A,B,A),aTP_Lamp_kj(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_3904_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_3905_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G3: fun(B,nat),P: product_prod(A,B)] : basic_BNF_size_prod(A,B,F3,G3,P) = 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,F3,aa(product_prod(A,B),A,product_fst(A,B),P))),aa(B,nat,G3,aa(product_prod(A,B),B,product_snd(A,B),P)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_3906_in__set__enumerate__eq,axiom,
    ! [A: $tType,P: product_prod(nat,A),Nb: nat,Xs: list(A)] :
      ( member(product_prod(nat,A),P,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Nb,Xs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(product_prod(nat,A),nat,product_fst(nat,A),P))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P)),Nb)) = aa(product_prod(nat,A),A,product_snd(nat,A),P) ) ) ) ).

% in_set_enumerate_eq
tff(fact_3907_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o)),Yb: A,Xb: B] :
      ( aa(B,$o,aa(A,fun(B,$o),P2,Yb),Xb)
     => aa(B,$o,aa(A,fun(B,$o),P2,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Yb))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Yb))) ) ).

% exI_realizer
tff(fact_3908_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),P: A,Q: fun(B,$o),Q2: B] :
      ( aa(A,$o,P2,P)
     => ( aa(B,$o,Q,Q2)
       => ( aa(A,$o,P2,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),P),Q2)))
          & aa(B,$o,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_3909_length__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,Nb,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_3910_distinct__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : distinct(product_prod(nat,A),enumerate(A,Nb,Xs)) ).

% distinct_enumerate
tff(fact_3911_bezw__non__0,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Yb)
     => ( bezw(Xb,Yb) = 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(Yb,modulo_modulo(nat,Xb,Yb)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Yb,modulo_modulo(nat,Xb,Yb)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Yb,modulo_modulo(nat,Xb,Yb)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xb,Yb))))) ) ) ).

% bezw_non_0
tff(fact_3912_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Yb )
     => ( Yb = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xb,Xaa)))))) ) ) ).

% bezw.elims
tff(fact_3913_bezw_Osimps,axiom,
    ! [Xb: nat,Yb: nat] :
      bezw(Xb,Yb) = $ite(Yb = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Yb,modulo_modulo(nat,Xb,Yb)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Yb,modulo_modulo(nat,Xb,Yb)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Yb,modulo_modulo(nat,Xb,Yb)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xb,Yb)))))) ).

% bezw.simps
tff(fact_3914_nth__enumerate__eq,axiom,
    ! [A: $tType,Mb: nat,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,Nb,Xs)),Mb) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)),aa(nat,A,nth(A,Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_3915_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Yb )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Yb = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xb,Xaa)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_3916_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A5: set(A),R2: A,S: A] :
          ( strict_mono_on(A,B,F3,A5)
         => ( member(A,R2,A5)
           => ( member(A,S,A5)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R2),S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R2)),aa(A,B,F3,S)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_3917_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [R3: A,S3: A] :
              ( member(A,R3,A5)
             => ( member(A,S3,A5)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R3),S3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R3)),aa(A,B,F3,S3)) ) ) )
         => strict_mono_on(A,B,F3,A5) ) ) ).

% strict_mono_onI
tff(fact_3918_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( strict_mono_on(A,B,F3,A5)
        <=> ! [R5: A,S7: A] :
              ( ( member(A,R5,A5)
                & member(A,S7,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R5),S7) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R5)),aa(A,B,F3,S7)) ) ) ) ).

% strict_mono_on_def
tff(fact_3919_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F3: fun(A,B),A5: set(A),Xb: A,Yb: A] :
          ( strict_mono_on(A,B,F3,A5)
         => ( member(A,Xb,A5)
           => ( member(A,Yb,A5)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb)) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_3920_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Yb )
     => ( accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Yb = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xb),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xb)))) )
           => ~ accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_3921_normalize__def,axiom,
    ! [P: product_prod(int,int)] :
      normalize(P) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P)),
        $let(
          a2: int,
          a2:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P)),aa(product_prod(int,int),int,product_snd(int,int),P)),
          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),P),a2)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P),a2)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P) = zero_zero(int),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
          $let(
            a2: int,
            a2:= aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P)),aa(product_prod(int,int),int,product_snd(int,int),P))),
            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),P),a2)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P),a2)) ) ) ) ).

% normalize_def
tff(fact_3922_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,Xb,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)),insert(A),Xb),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_3923_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),A2,aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_3924_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_3925_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Mb),Nb)),Nb) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add1
tff(fact_3926_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add2
tff(fact_3927_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_3928_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_3929_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Mb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),A2) ) ).

% drop_bit_drop_bit
tff(fact_3930_in__set__remove1,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( ( A2 != B2 )
     => ( member(A,A2,aa(list(A),set(A),set2(A),remove1(A,B2,Xs)))
      <=> member(A,A2,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% in_set_remove1
tff(fact_3931_gcd__pos__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Mb),Nb))
    <=> ( ( Mb != zero_zero(int) )
        | ( Nb != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_3932_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,B2: $o] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa($o,A,zero_neq_one_of_bool(A),(B2))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Nb = zero_zero(nat) )
            & (B2) )) ) ).

% drop_bit_of_bool
tff(fact_3933_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),Nb = zero_zero(nat)) ).

% drop_bit_of_Suc_0
tff(fact_3934_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_3935_remove1__commute,axiom,
    ! [A: $tType,Xb: A,Yb: A,Zs: list(A)] : remove1(A,Xb,remove1(A,Yb,Zs)) = remove1(A,Yb,remove1(A,Xb,Zs)) ).

% remove1_commute
tff(fact_3936_remove1__idem,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( remove1(A,Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_3937_notin__set__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: A] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ~ member(A,Xb,aa(list(A),set(A),set2(A),remove1(A,Yb,Xs))) ) ).

% notin_set_remove1
tff(fact_3938_distinct__remove1,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => distinct(A,remove1(A,Xb,Xs)) ) ).

% distinct_remove1
tff(fact_3939_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,K2: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K2),Mb)),Nb)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add_mult
tff(fact_3940_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = A2 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_3941_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),A2)) ) ).

% take_bit_drop_bit
tff(fact_3942_set__remove1__subset,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,Xb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_remove1_subset
tff(fact_3943_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),A2) ) ).

% gcd_le1_int
tff(fact_3944_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),B2) ) ).

% gcd_le2_int
tff(fact_3945_gcd__non__0__int,axiom,
    ! [Yb: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Yb)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Yb) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Yb),modulo_modulo(int,Xb,Yb)) ) ) ).

% gcd_non_0_int
tff(fact_3946_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = A2 ) ).

% bits_ident
tff(fact_3947_length__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),remove1(A,Xb,Xs)) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_remove1
tff(fact_3948_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),Xb) = powr(real,Xb,divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_3949_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or5935395276787703475ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_3950_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xb: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xb,B2)
         => ( ( B2 != zero_zero(B) )
           => ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A2,B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_3951_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xb: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xb,B2)
         => ( ( B2 != zero_zero(B) )
           => ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A2,B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_3952_gcd__0__left__nat,axiom,
    ! [Xb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),Xb) = Xb ).

% gcd_0_left_nat
tff(fact_3953_gcd__0__nat,axiom,
    ! [Xb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),zero_zero(nat)) = Xb ).

% gcd_0_nat
tff(fact_3954_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_3955_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_3956_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_3957_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_3958_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_3959_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb))
    <=> ( ( Mb != zero_zero(nat) )
        | ( Nb != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_3960_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( member(A,I2,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_3961_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),$o,finite_finite2(int),set_or5935395276787703475ssThan(int,L,U)) ).

% finite_greaterThanLessThan_int
tff(fact_3962_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K2)
         => ( set_or5935395276787703475ssThan(A,K2,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_3963_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_3964_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A2,B2) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_3965_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_3966_real__root__Suc__0,axiom,
    ! [Xb: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),Xb) = Xb ).

% real_root_Suc_0
tff(fact_3967_root__0,axiom,
    ! [Xb: real] : aa(real,real,root(zero_zero(nat)),Xb) = zero_zero(real) ).

% root_0
tff(fact_3968_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = aa(real,real,root(Nb),Yb) )
      <=> ( Xb = Yb ) ) ) ).

% real_root_eq_iff
tff(fact_3969_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_3970_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Yb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb) ) ) ).

% real_root_less_iff
tff(fact_3971_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Yb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Yb) ) ) ).

% real_root_le_iff
tff(fact_3972_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = one_one(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_3973_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_3974_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_3975_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Nb),Yb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Yb) ) ) ).

% real_root_gt_0_iff
tff(fact_3976_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_3977_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Yb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Yb) ) ) ).

% real_root_ge_0_iff
tff(fact_3978_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_3979_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,root(Nb),Yb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Yb) ) ) ).

% real_root_gt_1_iff
tff(fact_3980_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_3981_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,root(Nb),Yb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Yb) ) ) ).

% real_root_ge_1_iff
tff(fact_3982_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_3983_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_3984_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2) ) ).

% gcd_le1_nat
tff(fact_3985_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_3986_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_3987_gcd__non__0__nat,axiom,
    ! [Yb: nat,Xb: nat] :
      ( ( Yb != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Yb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Yb),modulo_modulo(nat,Xb,Yb)) ) ) ).

% gcd_non_0_nat
tff(fact_3988_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Yb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Yb) = $ite(Yb = zero_zero(nat),Xb,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Yb),modulo_modulo(nat,Xb,Yb))) ).

% gcd_nat.simps
tff(fact_3989_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Xaa) = Yb )
     => ( Yb = $ite(Xaa = zero_zero(nat),Xb,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xb,Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_3990_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A2,B2)) ) ) ).

% infinite_Ioo
tff(fact_3991_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Yb)) ) ) ).

% real_root_less_mono
tff(fact_3992_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Yb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Yb)) ) ) ).

% real_root_le_mono
tff(fact_3993_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(nat,real,power_power(real,Xb),K2)) = aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),K2) ) ) ).

% real_root_power
tff(fact_3994_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(real,real,abs_abs(real),Xb)) = aa(real,real,abs_abs(real),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_3995_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X3: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) ) ).

% bezout_nat
tff(fact_3996_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X3: nat,Y: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_3997_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( sgn_sgn(real,aa(real,real,root(Nb),Xb)) = sgn_sgn(real,Xb) ) ) ).

% sgn_root
tff(fact_3998_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_3999_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_4000_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_4001_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N3: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(N3),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4002_root__abs__power,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Nb),aa(nat,real,power_power(real,Yb),Nb))) = aa(real,real,abs_abs(real),Yb) ) ) ).

% root_abs_power
tff(fact_4003_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Yb: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,Xb,Yb),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Yb)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_4004_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4005_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_4006_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Yb: A,A2: B,B2: B,C3: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Yb),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C3)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4007_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Yb: A,C3: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Yb),C3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C3)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C3)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4008_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Yb: A,C3: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Yb),C3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C3) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C3)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C3)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4009_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Yb: A,A2: B,B2: B,C3: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Yb),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C3)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C3)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4010_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)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_4011_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_4012_real__root__strict__increasing,axiom,
    ! [Nb: nat,N3: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N3),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4013_real__root__decreasing,axiom,
    ! [Nb: nat,N3: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N3),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_4014_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_4015_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),aa(nat,real,power_power(real,Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_4016_real__root__pos__unique,axiom,
    ! [Nb: nat,Yb: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Yb)
       => ( ( aa(nat,real,power_power(real,Yb),Nb) = Xb )
         => ( aa(real,real,root(Nb),Xb) = Yb ) ) ) ) ).

% real_root_pos_unique
tff(fact_4017_real__root__increasing,axiom,
    ! [Nb: nat,N3: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N3),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_4018_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Yb: A,C3: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Yb),C3)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_4019_root__sgn__power,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Yb)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Yb)),Nb))) = Yb ) ) ).

% root_sgn_power
tff(fact_4020_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,aa(real,real,root(Nb),Xb))),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),aa(real,real,root(Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_4021_ln__root,axiom,
    ! [Nb: nat,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Nb),B2)) = divide_divide(real,aa(real,real,ln_ln(real),B2),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% ln_root
tff(fact_4022_log__root,axiom,
    ! [Nb: nat,A2: real,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( aa(real,real,log(B2),aa(real,real,root(Nb),A2)) = divide_divide(real,aa(real,real,log(B2),A2),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_root
tff(fact_4023_log__base__root,axiom,
    ! [Nb: nat,B2: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,log(aa(real,real,root(Nb),B2)),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Xb)) ) ) ) ).

% log_base_root
tff(fact_4024_split__root,axiom,
    ! [P2: fun(real,$o),Nb: nat,Xb: real] :
      ( aa(real,$o,P2,aa(real,real,root(Nb),Xb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(real,$o,P2,zero_zero(real)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ! [Y4: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y4)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Y4)),Nb)) = Xb )
             => aa(real,$o,P2,Y4) ) ) ) ) ).

% split_root
tff(fact_4025_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Xaa) = Yb )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Yb = $ite(Xaa = zero_zero(nat),Xb,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xb,Xaa))) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_4026_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,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_kk(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_4027_card__UNION,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A5)
     => ( ! [X3: set(A)] :
            ( member(set(A),X3,A5)
           => aa(set(A),$o,finite_finite2(A),X3) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_kl(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_km(set(set(A)),fun(set(set(A)),$o),A5)))) ) ) ) ).

% card_UNION
tff(fact_4028_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ai(nat,fun(nat,$o)),aTP_Lamp_ah(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_4029_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or5935395276787703475ssThan(nat,L,U)) ).

% finite_greaterThanLessThan
tff(fact_4030_Suc__funpow,axiom,
    ! [Nb: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),Nb),suc) = aa(nat,fun(nat,nat),plus_plus(nat),Nb) ).

% Suc_funpow
tff(fact_4031_Sup__lessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Yb: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_lessThan(A),Yb)) = Yb ) ).

% Sup_lessThan
tff(fact_4032_finite__Inter,axiom,
    ! [A: $tType,M4: set(set(A))] :
      ( ? [X: set(A)] :
          ( member(set(A),X,M4)
          & aa(set(A),$o,finite_finite2(A),X) )
     => aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),M4)) ) ).

% finite_Inter
tff(fact_4033_Sup__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Yb: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atMost(A),Yb)) = Yb ) ).

% Sup_atMost
tff(fact_4034_funpow__0,axiom,
    ! [A: $tType,F3: fun(A,A),Xb: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3),Xb) = Xb ).

% funpow_0
tff(fact_4035_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Yb,Xb)) = Xb ) ) ) ).

% cSup_atLeastAtMost
tff(fact_4036_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Xb,Yb)) = Yb ) ) ) ).

% Sup_atLeastAtMost
tff(fact_4037_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = Xb ) ).

% cSup_singleton
tff(fact_4038_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Yb,Xb)) = Yb ) ) ) ).

% cInf_atLeastAtMost
tff(fact_4039_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Xb,Yb)) = Xb ) ) ) ).

% Inf_atLeastAtMost
tff(fact_4040_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = Xb ) ).

% cInf_singleton
tff(fact_4041_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Xb,Yb)) = Yb ) ) ) ).

% Sup_atLeastLessThan
tff(fact_4042_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Yb,Xb)) = Xb ) ) ) ).

% cSup_atLeastLessThan
tff(fact_4043_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Xb,Yb)) = Xb ) ) ) ).

% Inf_atLeastLessThan
tff(fact_4044_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Yb,Xb)) = Yb ) ) ) ).

% cInf_atLeastLessThan
tff(fact_4045_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atMost(A),Xb)) = bot_bot(A) ) ).

% Inf_atMost
tff(fact_4046_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Yb,Xb)) = Xb ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_4047_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Xb,Yb)) = Yb ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_4048_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Xb,Yb)) = Xb ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_4049_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Yb,Xb)) = Yb ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_4050_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),aa(nat,nat,suc,L)) ).

% card_greaterThanLessThan
tff(fact_4051_finite__Union,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A5)
     => ( ! [M7: set(A)] :
            ( member(set(A),M7,A5)
           => aa(set(A),$o,finite_finite2(A),M7) )
       => aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)) ) ) ).

% finite_Union
tff(fact_4052_funpow__mult,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F3)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),F3) ).

% funpow_mult
tff(fact_4053_funpow__swap1,axiom,
    ! [A: $tType,F3: fun(A,A),Nb: nat,Xb: A] : aa(A,A,F3,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),Xb)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),aa(A,A,F3,Xb)) ).

% funpow_swap1
tff(fact_4054_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semila1105856199041335345_order(A,F3,Z,Less_eq,Less)
     => ( ( Z = aa(A,A,aa(A,fun(A,A),F3,A2),B2) )
      <=> ( ( A2 = Z )
          & ( B2 = Z ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
tff(fact_4055_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A2: A,B2: A] :
      ( semila1105856199041335345_order(A,F3,Z,Less_eq,Less)
     => ( ( aa(A,A,aa(A,fun(A,A),F3,A2),B2) = Z )
      <=> ( ( A2 = Z )
          & ( B2 = Z ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
tff(fact_4056_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X5: set(A)] :
          ( member(A,Z,X5)
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_4057_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X5: set(A),A2: A] :
          ( ! [X3: A] :
              ( member(A,X3,X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A2) )
         => ( ! [Y: A] :
                ( ! [X: A] :
                    ( member(A,X,X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_4058_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X5: set(A)] :
          ( member(A,Z,X5)
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X3) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_4059_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X5: set(A),A2: A] :
          ( ! [X3: A] :
              ( member(A,X3,X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3) )
         => ( ! [Y: A] :
                ( ! [X: A] :
                    ( member(A,X,X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A2) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_4060_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3) )
           => ( ! [Y: A] :
                  ( ! [X: A] :
                      ( member(A,X,X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A2) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X5) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4061_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ).

% cInf_greatest
tff(fact_4062_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( member(A,Xb,X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Xb) ) ) ) ).

% cInf_le_finite
tff(fact_4063_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Z)
           => ? [X3: A] :
                ( member(A,X3,X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z) ) ) ) ) ).

% cInf_lessD
tff(fact_4064_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Xb: A,A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( member(A,Xb,X5)
           => ( ! [X3: A] :
                  ( member(A,X3,X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X3) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_4065_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A2) )
           => ( ! [Y: A] :
                  ( ! [X: A] :
                      ( member(A,X,X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_4066_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X5)),Z) ) ) ) ).

% cSup_least
tff(fact_4067_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( member(A,Xb,X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ).

% le_cSup_finite
tff(fact_4068_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X5))
           => ? [X3: A] :
                ( member(A,X3,X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3) ) ) ) ) ).

% less_cSupD
tff(fact_4069_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Yb: A,X5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),aa(set(A),A,complete_Sup_Sup(A),X5))
         => ( ( X5 != bot_bot(set(A)) )
           => ~ ! [X3: A] :
                  ( member(A,X3,X5)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),X3) ) ) ) ) ).

% less_cSupE
tff(fact_4070_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Xb: A,A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( member(A,Xb,X5)
           => ( ! [X3: A] :
                  ( member(A,X3,X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),A2) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A2) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_4071_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2)) ).

% card_Union_le_sum_card
tff(fact_4072_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_4073_finite__UnionD,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5))
     => aa(set(set(A)),$o,finite_finite2(set(A)),A5) ) ).

% finite_UnionD
tff(fact_4074_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X5))
            <=> ! [X4: A] :
                  ( member(A,X4,X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_4075_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A2)
            <=> ! [X4: A] :
                  ( member(A,X4,X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A2) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_4076_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S2))),A2) ) ) ) ).

% cInf_abs_ge
tff(fact_4077_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S2))),A2) ) ) ) ).

% cSup_abs_le
tff(fact_4078_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X3: set(A)] :
          ( member(set(A),X3,U2)
         => aa(set(A),$o,finite_finite2(A),X3) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2)) ) ).

% card_Union_le_sum_card_weak
tff(fact_4079_finite__subset__Union,axiom,
    ! [A: $tType,A5: set(A),B11: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11))
       => ~ ! [F6: set(set(A))] :
              ( aa(set(set(A)),$o,finite_finite2(set(A)),F6)
             => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),F6),B11)
               => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F6)) ) ) ) ) ).

% finite_subset_Union
tff(fact_4080_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),Nb) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_4081_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K2: num,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K2)),A2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K2)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A2) ) ).

% numeral_add_unfold_funpow
tff(fact_4082_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E3: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S2)),L))),E3) ) ) ) ).

% cInf_asclose
tff(fact_4083_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E3: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S2)),L))),E3) ) ) ) ).

% cSup_asclose
tff(fact_4084_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_kn(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_4085_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),S2)) = $ite(S2 = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,complete_Sup_Sup(A),S2))) ) ) ) ).

% Sup_insert_finite
tff(fact_4086_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K2: num] : aa(num,A,numeral_numeral(A),K2) = 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),K2)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_4087_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xb: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = Xb ) ).

% ccpo_Sup_singleton
tff(fact_4088_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_4089_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_4090_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A5) = bot_bot(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X4)
             => ? [Xa2: A] :
                  ( member(A,Xa2,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X4) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_4091_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A5) = bot_bot(A) )
        <=> ! [X4: A] :
              ( member(A,X4,A5)
             => ( X4 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_4092_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A5) )
        <=> ! [X4: A] :
              ( member(A,X4,A5)
             => ( X4 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_4093_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_4094_Inf__nat__def1,axiom,
    ! [K5: set(nat)] :
      ( ( K5 != bot_bot(set(nat)) )
     => member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K5),K5) ) ).

% Inf_nat_def1
tff(fact_4095_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Xb: A] :
          ( ! [Y: A] :
              ( member(A,Y,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) )
         => ( ! [Y: A] :
                ( ! [Z3: A] :
                    ( member(A,Z3,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),Y) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A5) = Xb ) ) ) ) ).

% Sup_eqI
tff(fact_4096_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ! [A4: A] :
              ( member(A,A4,A5)
             => ? [X: A] :
                  ( member(A,X,B5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ).

% Sup_mono
tff(fact_4097_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Z: A] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z) ) ) ).

% Sup_least
tff(fact_4098_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A5: set(A)] :
          ( member(A,Xb,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ).

% Sup_upper
tff(fact_4099_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B2)
        <=> ! [X4: A] :
              ( member(A,X4,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B2) ) ) ) ).

% Sup_le_iff
tff(fact_4100_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A5: set(A),V: A] :
          ( member(A,U,A5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V),U)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ).

% Sup_upper2
tff(fact_4101_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S2: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2))
        <=> ? [X4: A] :
              ( member(A,X4,S2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4) ) ) ) ).

% less_Sup_iff
tff(fact_4102_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Xb: A] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),I3) )
         => ( ! [Y: A] :
                ( ! [I: A] :
                    ( member(A,I,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),I) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A5) = Xb ) ) ) ) ).

% Inf_eqI
tff(fact_4103_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( ! [B4: A] :
              ( member(A,B4,B5)
             => ? [X: A] :
                  ( member(A,X,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B4) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ).

% Inf_mono
tff(fact_4104_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A5: set(A)] :
          ( member(A,Xb,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),Xb) ) ) ).

% Inf_lower
tff(fact_4105_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A5: set(A),V: A] :
          ( member(A,U,A5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),V) ) ) ) ).

% Inf_lower2
tff(fact_4106_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A5))
        <=> ! [X4: A] :
              ( member(A,X4,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X4) ) ) ) ).

% le_Inf_iff
tff(fact_4107_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Z: A] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A5)) ) ) ).

% Inf_greatest
tff(fact_4108_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S2: set(A),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2)
        <=> ? [X4: A] :
              ( member(A,X4,S2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A2) ) ) ) ).

% Inf_less_iff
tff(fact_4109_Union__empty__conv,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5) = bot_bot(set(A)) )
    <=> ! [X4: set(A)] :
          ( member(set(A),X4,A5)
         => ( X4 = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_4110_empty__Union__conv,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5) )
    <=> ! [X4: set(A)] :
          ( member(set(A),X4,A5)
         => ( X4 = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_4111_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,A5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A5))
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xb)
             => ? [X4: A] :
                  ( member(A,X4,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X4) ) ) ) ) ).

% le_Sup_iff
tff(fact_4112_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),Xb)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y4)
             => ? [X4: A] :
                  ( member(A,X4,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4) ) ) ) ) ).

% Inf_le_iff
tff(fact_4113_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3) )
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ).

% less_eq_Sup
tff(fact_4114_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ).

% Sup_subset_mono
tff(fact_4115_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),U) )
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),U) ) ) ) ).

% Inf_less_eq
tff(fact_4116_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ).

% Inf_superset_mono
tff(fact_4117_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_4118_Inter__subset,axiom,
    ! [A: $tType,A5: set(set(A)),B5: set(A)] :
      ( ! [X7: set(A)] :
          ( member(set(A),X7,A5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B5) )
     => ( ( A5 != bot_bot(set(set(A))) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),B5) ) ) ).

% Inter_subset
tff(fact_4119_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ).

% Inf_le_Sup
tff(fact_4120_relpowp__bot,axiom,
    ! [A: $tType,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),bot_bot(fun(A,fun(A,$o)))) = bot_bot(fun(A,fun(A,$o))) ) ) ).

% relpowp_bot
tff(fact_4121_relpowp__fun__conv,axiom,
    ! [A: $tType,Nb: nat,P2: fun(A,fun(A,$o)),Xb: A,Yb: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P2),Xb),Yb)
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = Xb )
          & ( aa(nat,A,F7,Nb) = Yb )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),P2,aa(nat,A,F7,I4)),aa(nat,A,F7,aa(nat,nat,suc,I4))) ) ) ) ).

% relpowp_fun_conv
tff(fact_4122_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_4123_card__partition,axiom,
    ! [A: $tType,C5: set(set(A)),K2: nat] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),C5)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5))
       => ( ! [C2: set(A)] :
              ( member(set(A),C2,C5)
             => ( aa(set(A),nat,finite_card(A),C2) = K2 ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( member(set(A),C1,C5)
               => ( member(set(A),C22,C5)
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(set(set(A)),nat,finite_card(set(A)),C5)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)) ) ) ) ) ) ).

% card_partition
tff(fact_4124_inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_inf(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xb: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),inf_inf(fun(B,A)),F3),G3),Xb) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F3,Xb)),aa(B,A,G3,Xb)) ) ).

% inf_apply
tff(fact_4125_inf__right__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),Yb) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) ) ).

% inf_right_idem
tff(fact_4126_inf_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(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),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ).

% inf.right_idem
tff(fact_4127_inf__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) ) ).

% inf_left_idem
tff(fact_4128_inf_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ).

% inf.left_idem
tff(fact_4129_inf__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Xb) = Xb ) ).

% inf_idem
tff(fact_4130_inf_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),A2) = A2 ) ).

% inf.idem
tff(fact_4131_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) ) ) ) ).

% le_inf_iff
tff(fact_4132_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% inf.bounded_iff
tff(fact_4133_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xb) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_4134_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_4135_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xb) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_left
tff(fact_4136_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),bot_bot(A)) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_right
tff(fact_4137_finite__Int,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( ( aa(set(A),$o,finite_finite2(A),F4)
        | aa(set(A),$o,finite_finite2(A),G5) )
     => aa(set(A),$o,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_4138_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,uminus_uminus(A),Xb)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_4139_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),Xb) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_4140_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),aa(A,A,uminus_uminus(A),Xb))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_4141_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),Yb)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_4142_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_4143_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)),B5) = bot_bot(set(A)) )
    <=> ( ~ member(A,A2,B5)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_4144_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: 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)),insert(A),A2),A5)),B5) )
    <=> ( ~ member(A,A2,B5)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ) ) ) ).

% insert_disjoint(2)
tff(fact_4145_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B5: set(A),A2: A,A5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)) = bot_bot(set(A)) )
    <=> ( ~ member(A,A2,B5)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A5) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_4146_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A5: set(A),B2: A,B5: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5)) )
    <=> ( ~ member(A,B2,A5)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ) ) ) ).

% disjoint_insert(2)
tff(fact_4147_Diff__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_4148_Compl__disjoint2,axiom,
    ! [A: $tType,A5: 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)),A5)),A5) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_4149_Compl__disjoint,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_4150_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A5: set(A),F3: fun(A,B),P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_ko(fun(A,B),fun(fun(A,$o),fun(A,B)),F3),P2)),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_4151_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A5: set(A),P2: fun(A,$o),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_kp(fun(A,$o),fun(fun(A,B),fun(A,B)),P2),F3)),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_4152_sum__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A5: set(A),P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_kq(fun(A,$o),fun(A,B),P2)),A5) = aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_4153_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ).

% Sup_inter_less_eq
tff(fact_4154_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( member(A,X4,A5)
         => ! [Xa2: A] :
              ( member(A,Xa2,B5)
             => ( X4 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4155_Int__empty__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_4156_Int__empty__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B5) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_4157_disjoint__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> ! [X4: A] :
          ( member(A,X4,A5)
         => ~ member(A,X4,B5) ) ) ).

% disjoint_iff
tff(fact_4158_Int__emptyI,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ! [X3: A] :
          ( member(A,X3,A5)
         => ~ member(A,X3,B5) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_4159_inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F3),G3),X) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,X)),aa(A,B,G3,X)) ) ).

% inf_fun_def
tff(fact_4160_inf__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z)) ) ).

% inf_left_commute
tff(fact_4161_inf_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A,C3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),C3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)) ) ).

% inf.left_commute
tff(fact_4162_inf__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Xb) ) ).

% inf_commute
tff(fact_4163_inf_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: 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),B2),A2) ) ).

% inf.commute
tff(fact_4164_inf__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z)) ) ).

% inf_assoc
tff(fact_4165_inf_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)) ) ).

% inf.assoc
tff(fact_4166_inf__sup__aci_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Xb) ) ).

% inf_sup_aci(1)
tff(fact_4167_inf__sup__aci_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z)) ) ).

% inf_sup_aci(2)
tff(fact_4168_inf__sup__aci_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z)) ) ).

% inf_sup_aci(3)
tff(fact_4169_inf__sup__aci_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) ) ).

% inf_sup_aci(4)
tff(fact_4170_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% less_infI1
tff(fact_4171_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xb: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% less_infI2
tff(fact_4172_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb3
tff(fact_4173_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_4174_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% inf.strict_boundedE
tff(fact_4175_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_4176_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.strict_coboundedI1
tff(fact_4177_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.strict_coboundedI2
tff(fact_4178_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.coboundedI2
tff(fact_4179_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.coboundedI1
tff(fact_4180_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_4181_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_4182_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),B2) ) ).

% inf.cobounded2
tff(fact_4183_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),A2) ) ).

% inf.cobounded1
tff(fact_4184_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_4185_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z)) ) ) ) ).

% inf_greatest
tff(fact_4186_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)) ) ) ) ).

% inf.boundedI
tff(fact_4187_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% inf.boundedE
tff(fact_4188_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = Yb ) ) ) ).

% inf_absorb2
tff(fact_4189_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = Xb ) ) ) ).

% inf_absorb1
tff(fact_4190_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_4191_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_4192_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = Xb ) ) ) ).

% le_iff_inf
tff(fact_4193_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F3: fun(A,fun(A,A)),Xb: A,Yb: A] :
          ( ! [X3: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y)),X3)
         => ( ! [X3: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y)),Y)
           => ( ! [X3: A,Y: A,Z2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,Y),Z2)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = aa(A,A,aa(A,fun(A,A),F3,Xb),Yb) ) ) ) ) ) ).

% inf_unique
tff(fact_4194_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% inf.orderI
tff(fact_4195_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_4196_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xb: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% le_infI2
tff(fact_4197_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% le_infI1
tff(fact_4198_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),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),C3),D3)) ) ) ) ).

% inf_mono
tff(fact_4199_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% le_infI
tff(fact_4200_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2) ) ) ) ).

% le_infE
tff(fact_4201_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),Yb) ) ).

% inf_le2
tff(fact_4202_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),Xb) ) ).

% inf_le1
tff(fact_4203_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),Xb) ) ).

% inf_sup_ord(1)
tff(fact_4204_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),Yb) ) ).

% inf_sup_ord(2)
tff(fact_4205_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: 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),Xb)),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),B2)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_4206_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: 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),Xb),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xb)),B2)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_4207_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B5)),A2) = bot_bot(A) )
        <=> ! [X4: A] :
              ( member(A,X4,B5)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X4),A2) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_4208_Int__Diff__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_4209_Diff__triv,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = A5 ) ) ).

% Diff_triv
tff(fact_4210_Union__disjoint,axiom,
    ! [A: $tType,C5: set(set(A)),A5: 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)),C5)),A5) = bot_bot(set(A)) )
    <=> ! [X4: set(A)] :
          ( member(set(A),X4,C5)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),A5) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_4211_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,Mb: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_4212_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Yb)) ) ) ).

% inf_shunt
tff(fact_4213_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] :
                  ( member(A,X3,A5)
                 => ( member(A,Y,A5)
                   => member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y),A5) ) )
             => member(A,aa(set(A),A,complete_Inf_Inf(A),A5),A5) ) ) ) ) ).

% finite_Inf_in
tff(fact_4214_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,Mb: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_4215_ivl__disj__int__one_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(4)
tff(fact_4216_ivl__disj__int__one_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(2)
tff(fact_4217_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ) ).

% disjoint_eq_subset_Compl
tff(fact_4218_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,Mb: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(5)
tff(fact_4219_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,Mb: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or5935395276787703475ssThan(A,Mb,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(4)
tff(fact_4220_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,Mb: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(1)
tff(fact_4221_ivl__disj__int__one_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(1)
tff(fact_4222_relpowp__0__I,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xb: A] : aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P2),Xb),Xb) ).

% relpowp_0_I
tff(fact_4223_relpowp__0__E,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xb: A,Yb: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P2),Xb),Yb)
     => ( Xb = Yb ) ) ).

% relpowp_0_E
tff(fact_4224_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),R) = fequal(A) ).

% relpowp.simps(1)
tff(fact_4225_insert__partition,axiom,
    ! [A: $tType,Xb: set(A),F4: set(set(A))] :
      ( ~ member(set(A),Xb,F4)
     => ( ! [X3: set(A)] :
            ( member(set(A),X3,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Xb),F4))
           => ! [Xa3: set(A)] :
                ( member(set(A),Xa3,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Xb),F4))
               => ( ( X3 != Xa3 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa3) = bot_bot(set(A)) ) ) ) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Xb),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F4)) = bot_bot(set(A)) ) ) ) ).

% insert_partition
tff(fact_4226_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(A),fun(A,B),aTP_Lamp_kr(fun(A,B),fun(set(A),fun(A,B)),G3),B5)),A5) ) ) ) ).

% sum.inter_restrict
tff(fact_4227_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_ks(fun(A,B),fun(set(A),fun(A,B)),G3),B5)),A5) ) ) ) ).

% prod.inter_restrict
tff(fact_4228_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S2: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( ! [I3: A] :
                  ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,H,I3) = zero_zero(B) ) )
             => ( ! [I3: A] :
                    ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T2))
                   => ( aa(A,B,G3,I3) = zero_zero(B) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T2))
                     => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T2) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_4229_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K2: A,Xb: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),K2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_4230_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ) ) ) ).

% sum.Int_Diff
tff(fact_4231_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ) ) ) ).

% prod.Int_Diff
tff(fact_4232_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S2: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T2)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( ! [I3: A] :
                  ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S2))
                 => ( aa(A,B,H,I3) = one_one(B) ) )
             => ( ! [I3: A] :
                    ( member(A,I3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T2))
                   => ( aa(A,B,G3,I3) = one_one(B) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T2))
                     => ( aa(A,B,G3,X3) = aa(A,B,H,X3) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T2) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_4233_card__Diff__subset__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ).

% card_Diff_subset_Int
tff(fact_4234_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),P2: fun(A,$o),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_kt(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P2),H),G3)),A5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P2))))) ) ) ) ).

% sum.If_cases
tff(fact_4235_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),P2: fun(A,$o),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ku(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P2),H),G3)),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P2))))) ) ) ) ).

% prod.If_cases
tff(fact_4236_dvd__partition,axiom,
    ! [A: $tType,C5: set(set(A)),K2: nat] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5))
     => ( ! [X3: set(A)] :
            ( member(set(A),X3,C5)
           => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),aa(set(A),nat,finite_card(A),X3)) )
       => ( ! [X3: set(A)] :
              ( member(set(A),X3,C5)
             => ! [Xa3: set(A)] :
                  ( member(set(A),Xa3,C5)
                 => ( ( X3 != Xa3 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa3) = bot_bot(set(A)) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5))) ) ) ) ).

% dvd_partition
tff(fact_4237_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A5: set(A),F3: fun(A,B),B2: B] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(B,fun(A,B),aTP_Lamp_kv(fun(A,B),fun(B,fun(A,B)),F3),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_kw(fun(A,B),fun(B,fun(A,$o)),F3),B2))))),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_kx(fun(A,B),fun(B,fun(A,$o)),F3),B2)))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_4238_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys3: list(A)] :
            ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs2: list(A)] :
              ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys3 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_4239_relpowp__E,axiom,
    ! [A: $tType,Nb: nat,P2: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P2),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M),P2),Xb),Y)
               => ~ aa(A,$o,aa(A,fun(A,$o),P2,Y),Z) ) ) ) ) ).

% relpowp_E
tff(fact_4240_relpowp__E2,axiom,
    ! [A: $tType,Nb: nat,P2: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P2),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(A,$o,aa(A,fun(A,$o),P2,Xb),Y)
               => ~ aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M),P2),Y),Z) ) ) ) ) ).

% relpowp_E2
tff(fact_4241_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_4242_totally__bounded__Union,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [M4: set(set(A))] :
          ( aa(set(set(A)),$o,finite_finite2(set(A)),M4)
         => ( ! [S6: set(A)] :
                ( 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_4243_set__removeAll,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,Xb),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)),insert(A),Xb),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_4244_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_4245_removeAll__id,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(list(A),list(A),removeAll(A,Xb),Xs) = Xs ) ) ).

% removeAll_id
tff(fact_4246_totally__bounded__empty,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => topolo6688025880775521714ounded(A,bot_bot(set(A))) ) ).

% totally_bounded_empty
tff(fact_4247_finite__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),shuffles(A,Xs,Ys)) ).

% finite_shuffles
tff(fact_4248_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B)),X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),S2)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),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_4249_int_Oabs__induct,axiom,
    ! [P2: fun(int,$o),Xb: int] :
      ( ! [Y: product_prod(nat,nat)] : aa(int,$o,P2,aa(product_prod(nat,nat),int,abs_Integ,Y))
     => aa(int,$o,P2,Xb) ) ).

% int.abs_induct
tff(fact_4250_shuffles__commutes,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : shuffles(A,Xs,Ys) = shuffles(A,Ys,Xs) ).

% shuffles_commutes
tff(fact_4251_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y: nat] : Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),Y)) ).

% eq_Abs_Integ
tff(fact_4252_distinct__removeAll,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),removeAll(A,Xb),Xs)) ) ).

% distinct_removeAll
tff(fact_4253_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => ( aa(list(A),nat,size_size(list(A)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ) ) ).

% length_shuffles
tff(fact_4254_nat_Oabs__eq,axiom,
    ! [Xb: product_prod(nat,nat)] : aa(int,nat,nat2,aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),Xb) ).

% nat.abs_eq
tff(fact_4255_length__removeAll__less__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,Xb),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_removeAll_less_eq
tff(fact_4256_distinct__remove1__removeAll,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( remove1(A,Xb,Xs) = aa(list(A),list(A),removeAll(A,Xb),Xs) ) ) ).

% distinct_remove1_removeAll
tff(fact_4257_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_4258_int__def,axiom,
    ! [Nb: nat] : aa(nat,int,semiring_1_of_nat(int),Nb) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Nb),zero_zero(nat))) ).

% int_def
tff(fact_4259_length__removeAll__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,Xb),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_removeAll_less
tff(fact_4260_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)) )
         => ( member(list(A),Zs,shuffles(A,Xs,Ys))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_4261_uminus__int_Oabs__eq,axiom,
    ! [Xb: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_la(nat,fun(nat,product_prod(nat,nat)))),Xb)) ).

% uminus_int.abs_eq
tff(fact_4262_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_4263_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: product_prod(nat,nat)] : aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = 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_lb(nat,fun(nat,A))),Xb) ) ).

% of_int.abs_eq
tff(fact_4264_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_4265_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_4266_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_lh(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_4267_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_lj(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_4268_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),aa(list(list(A)),list(list(A)),removeAll(list(A),nil(A)),Xs))
        & ! [Ys4: list(A)] :
            ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys4) )
        & ! [Ys4: list(A),Zs3: list(A)] :
            ( ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & member(list(A),Zs3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & ( Ys4 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys4)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_4269_distinct__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ! [X3: list(A)] :
          ( member(list(A),X3,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => distinct(A,X3) )
     => distinct(list(A),product_lists(A,Xss)) ) ).

% distinct_product_lists
tff(fact_4270_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_4271_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_4272_list__update__nonempty,axiom,
    ! [A: $tType,Xs: list(A),K2: nat,Xb: A] :
      ( ( list_update(A,Xs,K2,Xb) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% list_update_nonempty
tff(fact_4273_concat__replicate__trivial,axiom,
    ! [A: $tType,I2: nat] : concat(A,replicate(list(A),I2,nil(A))) = nil(A) ).

% concat_replicate_trivial
tff(fact_4274_Nil__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( member(list(A),nil(A),shuffles(A,Xs,Ys))
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_in_shuffles
tff(fact_4275_enumerate__simps_I1_J,axiom,
    ! [A: $tType,Nb: nat] : enumerate(A,Nb,nil(A)) = nil(product_prod(nat,A)) ).

% enumerate_simps(1)
tff(fact_4276_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_4277_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_4278_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_4279_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_4280_empty__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( nil(A) = replicate(A,Nb,Xb) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_4281_replicate__empty,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( replicate(A,Nb,Xb) = nil(A) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_4282_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A] : groups4207007520872428315er_sum(B,A,F3,A2,nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_4283_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( nil(A) = concat(A,Xss) )
    <=> ! [X4: list(A)] :
          ( member(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X4 = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_4284_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( concat(A,Xss) = nil(A) )
    <=> ! [X4: list(A)] :
          ( member(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X4 = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_4285_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_4286_removeAll_Osimps_I1_J,axiom,
    ! [A: $tType,Xb: A] : aa(list(A),list(A),removeAll(A,Xb),nil(A)) = nil(A) ).

% removeAll.simps(1)
tff(fact_4287_Nil__in__shufflesI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = nil(A) )
     => ( ( Ys = nil(A) )
       => member(list(A),nil(A),shuffles(A,Xs,Ys)) ) ) ).

% Nil_in_shufflesI
tff(fact_4288_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))),insert(list(A)),Xs),bot_bot(set(list(A)))) ).

% shuffles.simps(2)
tff(fact_4289_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))),insert(list(A)),Ys),bot_bot(set(list(A)))) ).

% shuffles.simps(1)
tff(fact_4290_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : distinct(A,nil(A)) ).

% distinct.simps(1)
tff(fact_4291_rotate1_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rotate1(A),nil(A)) = nil(A) ).

% rotate1.simps(1)
tff(fact_4292_remove1_Osimps_I1_J,axiom,
    ! [A: $tType,Xb: A] : remove1(A,Xb,nil(A)) = nil(A) ).

% remove1.simps(1)
tff(fact_4293_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_4294_list__update__code_I1_J,axiom,
    ! [A: $tType,I2: nat,Yb: A] : list_update(A,nil(A),I2,Yb) = nil(A) ).

% list_update_code(1)
tff(fact_4295_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,I2: nat,V: A] : list_update(A,nil(A),I2,V) = nil(A) ).

% list_update.simps(1)
tff(fact_4296_concat_Osimps_I1_J,axiom,
    ! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).

% concat.simps(1)
tff(fact_4297_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_4298_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_4299_replicate__0,axiom,
    ! [A: $tType,Xb: A] : replicate(A,zero_zero(nat),Xb) = nil(A) ).

% replicate_0
tff(fact_4300_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : aa(list(A),nat,size_list(A,Xb),nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_4301_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Yb: A] : aa(A,nat,count_list(A,nil(A)),Yb) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_4302_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow2(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_4303_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( member(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Xss) ) ) ).

% in_set_product_lists_length
tff(fact_4304_less__eq__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,Xb)),aa(int,product_prod(nat,nat),rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_4305_less__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,Xb)),aa(int,product_prod(nat,nat),rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_4306_prod__encode__def,axiom,
    nat_prod_encode = aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),aTP_Lamp_lk(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_4307_Gcd__remove0__nat,axiom,
    ! [M4: set(nat)] :
      ( aa(set(nat),$o,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)),insert(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_4308_prod__encode__eq,axiom,
    ! [Xb: product_prod(nat,nat),Yb: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),nat,nat_prod_encode,Xb) = aa(product_prod(nat,nat),nat,nat_prod_encode,Yb) )
    <=> ( Xb = Yb ) ) ).

% prod_encode_eq
tff(fact_4309_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_4310_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)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ).

% Gcd_2
tff(fact_4311_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ( gcd_Gcd(A,A5) = zero_zero(A) )
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).

% Gcd_0_iff
tff(fact_4312_Gcd__in,axiom,
    ! [A5: set(nat)] :
      ( ! [A4: nat,B4: nat] :
          ( member(nat,A4,A5)
         => ( member(nat,B4,A5)
           => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B4),A5) ) )
     => ( ( A5 != bot_bot(set(nat)) )
       => member(nat,gcd_Gcd(nat,A5),A5) ) ) ).

% Gcd_in
tff(fact_4313_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),B2))) ).

% le_prod_encode_2
tff(fact_4314_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),B2))) ).

% le_prod_encode_1
tff(fact_4315_nat_Orep__eq,axiom,
    ! [Xb: int] : aa(int,nat,nat2,Xb) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),aa(int,product_prod(nat,nat),rep_Integ,Xb)) ).

% nat.rep_eq
tff(fact_4316_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int] : aa(int,A,ring_1_of_int(A),Xb) = 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_lb(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,Xb)) ) ).

% of_int.rep_eq
tff(fact_4317_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,Mb: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K2,Mb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K2)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_4318_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))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_4319_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ll(nat,$o))) ) ).

% semiring_char_def
tff(fact_4320_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_la(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_4321_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B2: A,P: product_prod(B,A)] :
      ( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P) )
    <=> ? [A7: B] : P = 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_4322_sum_Osize__neq,axiom,
    ! [A: $tType,B: $tType,Xb: sum_sum(A,B)] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),Xb) != zero_zero(nat) ).

% sum.size_neq
tff(fact_4323_prod_Osize__neq,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B)] : aa(product_prod(A,B),nat,size_size(product_prod(A,B)),Xb) != zero_zero(nat) ).

% prod.size_neq
tff(fact_4324_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A2: A,P: product_prod(A,B)] :
      ( ( A2 = aa(product_prod(A,B),A,product_fst(A,B),P) )
    <=> ? [B7: B] : P = 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_4325_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_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_4326_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_lj(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_4327_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_lh(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_4328_sndI,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),Yb: A,Z: B] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Yb),Z) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),Xb) = Z ) ) ).

% sndI
tff(fact_4329_fstI,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),Yb: A,Z: B] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Yb),Z) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),Xb) = Yb ) ) ).

% fstI
tff(fact_4330_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Nb,one2) ) ).

% add_neg_numeral_special(4)
tff(fact_4331_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,Mb,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_4332_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),one_one(A)) = neg_numeral_sub(A,one2,Mb) ) ).

% add_neg_numeral_special(2)
tff(fact_4333_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))) = neg_numeral_sub(A,one2,Mb) ) ).

% add_neg_numeral_special(1)
tff(fact_4334_sub__num__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_sub(A,one2,one2) = zero_zero(A) ) ) ).

% sub_num_simps(1)
tff(fact_4335_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W2: num,Yb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W2)),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W2,V)),Yb) ) ).

% semiring_norm(167)
tff(fact_4336_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W2: num,Yb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Yb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V,W2)),Yb) ) ).

% semiring_norm(166)
tff(fact_4337_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Mb,Nb) ) ).

% add_neg_numeral_simps(1)
tff(fact_4338_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Nb,Mb) ) ).

% add_neg_numeral_simps(2)
tff(fact_4339_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,Nb,Mb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ) ).

% sub_non_negative
tff(fact_4340_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),neg_numeral_sub(A,Nb,Mb)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),Mb) ) ) ).

% sub_non_positive
tff(fact_4341_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),neg_numeral_sub(A,Nb,Mb)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Nb),Mb) ) ) ).

% sub_negative
tff(fact_4342_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),neg_numeral_sub(A,Nb,Mb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ) ).

% sub_positive
tff(fact_4343_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),P: fun(A,B),I2: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(A,$o)),I6),P)))
         => ( groups1962203154675924110t_prod(A,B,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),I6)) = $ite(member(A,I2,I6),groups1962203154675924110t_prod(A,B,P,I6),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P,I2)),groups1962203154675924110t_prod(A,B,P,I6))) ) ) ) ).

% prod.insert'
tff(fact_4344_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( 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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))) = remove1(A,Xb,aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_4345_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C3: nat,Xb: nat,Yb: nat] :
      aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_lm(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,Xb,Yb)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C3),Yb),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Yb),C3)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Yb),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4346_rat__floor__lemma,axiom,
    ! [A2: int,B2: int] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),divide_divide(int,A2,B2))),fract(A2,B2))
      & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(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_4347_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),A5) = bot_bot(set(A)) )
    <=> ( A5 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_4348_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F3),A5) )
    <=> ( A5 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_4349_image__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] : aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_4350_finite__imageI,axiom,
    ! [B: $tType,A: $tType,F4: set(A),H: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,H),F4)) ) ).

% finite_imageI
tff(fact_4351_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_4352_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K2)),set_or1337092689740270186AtMost(A,I2,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2)) ) ).

% image_add_atLeastAtMost
tff(fact_4353_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),D3)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),A2)) ) ).

% image_diff_atLeastAtMost
tff(fact_4354_image__uminus__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Yb: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or1337092689740270186AtMost(A,Xb,Yb)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Yb),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_atLeastAtMost
tff(fact_4355_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K2)),set_or7035219750837199246ssThan(A,I2,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2)) ) ).

% image_add_atLeastLessThan
tff(fact_4356_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C3: A,A2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C3)),aa(A,set(A),set_ord_atMost(A),A2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)) ) ).

% image_add_atMost
tff(fact_4357_image__uminus__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Yb: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or5935395276787703475ssThan(A,Xb,Yb)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Yb),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_greaterThanLessThan
tff(fact_4358_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_4359_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_4360_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(list(A),set(A),set2(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) = A5 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_4361_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P: fun(B,A)] : groups1962203154675924110t_prod(B,A,P,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_4362_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(list(A),nat,size_size(list(A)),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) = aa(set(A),nat,finite_card(A),A5) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_4363_prod_Oeq__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),P: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( groups1962203154675924110t_prod(A,B,P,I6) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),P),I6) ) ) ) ).

% prod.eq_sum
tff(fact_4364_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,I2,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I2),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_4365_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,I2,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I2),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_4366_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: fun(B,A),A5: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B5),A5)) )
        <=> ! [X4: B] :
              ( member(B,X4,A5)
             => ( aa(B,A,B5,X4) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_4367_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B5),A5)) = bot_bot(A) )
        <=> ! [X4: B] :
              ( member(B,X4,A5)
             => ( aa(B,A,B5,X4) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_4368_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_ln(B,A)),A5)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_4369_ccSUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lo(B,A)),A5)) = bot_bot(A) ) ).

% ccSUP_bot
tff(fact_4370_SUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lp(B,fun(A,B),F3)),A5)) = F3 ) ) ) ).

% SUP_const
tff(fact_4371_ccSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),F3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lq(B,fun(A,B),F3)),A5)) = F3 ) ) ) ).

% ccSUP_const
tff(fact_4372_cSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),C3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lr(B,fun(A,B),C3)),A5)) = C3 ) ) ) ).

% cSUP_const
tff(fact_4373_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ls(A,fun(A,A),K2)),set_or1337092689740270186AtMost(A,I2,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2)) ) ).

% image_add_atLeastAtMost'
tff(fact_4374_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lp(B,fun(A,B),F3)),A5)) = F3 ) ) ) ).

% INF_const
tff(fact_4375_ccINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),F3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lq(B,fun(A,B),F3)),A5)) = F3 ) ) ) ).

% ccINF_const
tff(fact_4376_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),C3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lr(B,fun(A,B),C3)),A5)) = C3 ) ) ) ).

% cINF_const
tff(fact_4377_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_lt(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_4378_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ls(A,fun(A,A),K2)),set_or7035219750837199246ssThan(A,I2,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2)) ) ).

% image_add_atLeastLessThan'
tff(fact_4379_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = nil(A) )
          <=> ( A5 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4380_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)) = bot_bot(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X4)
             => ? [Xa2: B] :
                  ( member(B,Xa2,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,Xa2)),X4) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_4381_ccSUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B)))) = bot_bot(A) ) ).

% ccSUP_empty
tff(fact_4382_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4383_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_lu(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,divide_divide(A,A2,D3),divide_divide(A,B2,D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4384_less__rat,axiom,
    ! [B2: int,D3: int,A2: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A2,B2)),fract(C3,D3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))) ) ) ) ).

% less_rat
tff(fact_4385_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P2: fun(set(A),$o)] :
      ( ! [B10: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B10),aa(set(B),set(A),image2(B,A,F3),A5))
         => aa(set(A),$o,P2,B10) )
    <=> ! [B10: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B10),A5)
         => aa(set(A),$o,P2,aa(set(B),set(A),image2(B,A,F3),B10)) ) ) ).

% all_subset_image
tff(fact_4386_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,F3),A5))
       => ? [X3: A] :
            ( member(A,X3,A5)
            & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_lv(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A5),F3),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_4387_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = aa(set(A),list(A),linord4507533701916653071of_set(A),B5) )
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( A5 = B5 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_4388_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: 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_lw(fun(B,A),fun(B,set(A)),F3)),A5)) = aa(set(B),set(A),image2(B,A,F3),A5) ).

% UNION_singleton_eq_range
tff(fact_4389_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : distinct(A,aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_4390_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),F3: fun(A,C),G3: fun(B,C)] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => ? [X: B] :
                  ( member(B,X,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,I3)),aa(B,C,G3,X)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B5)
               => ? [X: A] :
                    ( member(A,X,A5)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G3,J2)),aa(A,C,F3,X)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,F3),A5)) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G3),B5)) ) ) ) ) ).

% SUP_eq
tff(fact_4391_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),G3: fun(B,C),F3: fun(A,C)] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => ? [X: B] :
                  ( member(B,X,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G3,X)),aa(A,C,F3,I3)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B5)
               => ? [X: A] :
                    ( member(A,X,A5)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,X)),aa(B,C,G3,J2)) ) )
           => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,F3),A5)) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G3),B5)) ) ) ) ) ).

% INF_eq
tff(fact_4392_zero__notin__Suc__image,axiom,
    ! [A5: set(nat)] : ~ member(nat,zero_zero(nat),aa(set(nat),set(nat),image2(nat,nat,suc),A5)) ).

% zero_notin_Suc_image
tff(fact_4393_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P2: fun(set(A),$o)] :
      ( ! [B10: set(A)] :
          ( ( aa(set(A),$o,finite_finite2(A),B10)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B10),aa(set(B),set(A),image2(B,A,F3),A5)) )
         => aa(set(A),$o,P2,B10) )
    <=> ! [B10: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),B10)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B10),A5) )
         => aa(set(A),$o,P2,aa(set(B),set(A),image2(B,A,F3),B10)) ) ) ).

% all_finite_subset_image
tff(fact_4394_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P2: fun(set(A),$o)] :
      ( ? [B10: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B10)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B10),aa(set(B),set(A),image2(B,A,F3),A5))
          & aa(set(A),$o,P2,B10) )
    <=> ? [B10: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),B10)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B10),A5)
          & aa(set(A),$o,P2,aa(set(B),set(A),image2(B,A,F3),B10)) ) ) ).

% ex_finite_subset_image
tff(fact_4395_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F3),A5))
       => ? [C6: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C6),A5)
            & aa(set(B),$o,finite_finite2(B),C6)
            & ( B5 = aa(set(B),set(A),image2(B,A,F3),C6) ) ) ) ) ).

% finite_subset_image
tff(fact_4396_finite__surj,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),aa(set(A),set(B),image2(A,B,F3),A5))
       => aa(set(B),$o,finite_finite2(B),B5) ) ) ).

% finite_surj
tff(fact_4397_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),Ta: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),S)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),Ta)) ) ).

% translation_Int
tff(fact_4398_SUP__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I6: set(A),F3: fun(A,B),Xb: B] :
          ( ( I6 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( member(A,I3,I6)
               => ( aa(A,B,F3,I3) = Xb ) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),I6)) = Xb ) ) ) ) ).

% SUP_eq_const
tff(fact_4399_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),Ta: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),S)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),Ta)) ) ).

% translation_diff
tff(fact_4400_INF__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I6: set(A),F3: fun(A,B),Xb: B] :
          ( ( I6 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( member(A,I3,I6)
               => ( aa(A,B,F3,I3) = Xb ) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),I6)) = Xb ) ) ) ) ).

% INF_eq_const
tff(fact_4401_finite__image__absD,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),image2(A,A,abs_abs(A)),S2))
         => aa(set(A),$o,finite_finite2(A),S2) ) ) ).

% finite_image_absD
tff(fact_4402_Rat__induct__pos,axiom,
    ! [P2: fun(rat,$o),Q2: rat] :
      ( ! [A4: int,B4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
         => aa(rat,$o,P2,fract(A4,B4)) )
     => aa(rat,$o,P2,Q2) ) ).

% Rat_induct_pos
tff(fact_4403_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),Ta)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),Ta)) ) ).

% translation_Compl
tff(fact_4404_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B),Xb: B] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),Xb) )
         => ( ! [Y: B] :
                ( ! [I: A] :
                    ( member(A,I,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),Y) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),Y) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5)) = Xb ) ) ) ) ).

% SUP_eqI
tff(fact_4405_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),F3: fun(A,C),G3: fun(B,C)] :
          ( ! [N2: A] :
              ( member(A,N2,A5)
             => ? [X: B] :
                  ( member(B,X,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,N2)),aa(B,C,G3,X)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,F3),A5))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G3),B5))) ) ) ).

% SUP_mono
tff(fact_4406_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B),U: B] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),U) ) ) ).

% SUP_least
tff(fact_4407_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F3: fun(A,B),G3: fun(A,B),A5: set(A)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G3),A5))) ) ) ).

% SUP_mono'
tff(fact_4408_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I2: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I2,A5)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ).

% SUP_upper
tff(fact_4409_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),U)
        <=> ! [X4: B] :
              ( member(B,X4,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X4)),U) ) ) ) ).

% SUP_le_iff
tff(fact_4410_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I2: A,A5: set(A),U: B,F3: fun(A,B)] :
          ( member(A,I2,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,I2))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ).

% SUP_upper2
tff(fact_4411_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)))
        <=> ? [X4: B] :
              ( member(B,X4,A5)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,X4)) ) ) ) ).

% less_SUP_iff
tff(fact_4412_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),Yb: A,I2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),Yb)
         => ( member(B,I2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,I2)),Yb) ) ) ) ).

% SUP_lessD
tff(fact_4413_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),Xb: B,F3: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),aa(A,B,F3,I3)) )
         => ( ! [Y: B] :
                ( ! [I: A] :
                    ( member(A,I,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),aa(A,B,F3,I)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Xb) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5)) = Xb ) ) ) ) ).

% INF_eqI
tff(fact_4414_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B5: set(A),A5: set(B),F3: fun(B,C),G3: fun(A,C)] :
          ( ! [M: A] :
              ( member(A,M,B5)
             => ? [X: B] :
                  ( member(B,X,A5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F3,X)),aa(A,C,G3,M)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F3),A5))),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,G3),B5))) ) ) ).

% INF_mono
tff(fact_4415_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I2: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I2,A5)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(A,B,F3,I2)) ) ) ).

% INF_lower
tff(fact_4416_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F3: fun(A,B),G3: fun(A,B),A5: set(A)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G3),A5))) ) ) ).

% INF_mono'
tff(fact_4417_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I2: A,A5: set(A),F3: fun(A,B),U: B] :
          ( member(A,I2,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),U) ) ) ) ).

% INF_lower2
tff(fact_4418_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)))
        <=> ! [X4: B] :
              ( member(B,X4,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F3,X4)) ) ) ) ).

% le_INF_iff
tff(fact_4419_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),U: B,F3: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ).

% INF_greatest
tff(fact_4420_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),A2)
        <=> ? [X4: B] :
              ( member(B,X4,A5)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X4)),A2) ) ) ) ).

% INF_less_iff
tff(fact_4421_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Yb: A,F3: fun(B,A),A5: set(B),I2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)))
         => ( member(B,I2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),aa(B,A,F3,I2)) ) ) ) ).

% less_INF_D
tff(fact_4422_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A5: set(A),F3: fun(nat,A),Nb: nat] :
      ( ( A5 = aa(set(nat),set(A),image2(nat,A,F3),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb))) )
     => aa(set(A),$o,finite_finite2(A),A5) ) ).

% nat_seg_image_imp_finite
tff(fact_4423_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
    <=> ? [N: nat,F7: fun(nat,A)] : A5 = aa(set(nat),set(A),image2(nat,A,F7),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),N))) ) ).

% finite_conv_nat_seg_image
tff(fact_4424_image__constant,axiom,
    ! [A: $tType,B: $tType,Xb: A,A5: set(A),C3: B] :
      ( member(A,Xb,A5)
     => ( aa(set(A),set(B),image2(A,B,aTP_Lamp_lx(B,fun(A,B),C3)),A5) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),C3),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_4425_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C3: A,A5: set(B)] :
      aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_kj(A,fun(B,A)),C3)),A5) = $ite(A5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C3),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_4426_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),H: fun(A,B),G3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),S2) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_lz(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H),G3)),aa(set(A),set(C),image2(A,C,G3),S2)) ) ) ) ).

% sum.image_gen
tff(fact_4427_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),H: fun(A,B),G3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S2) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ma(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H),G3)),aa(set(A),set(C),image2(A,C,G3),S2)) ) ) ) ).

% prod.image_gen
tff(fact_4428_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B),Xb: A] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ! [Y: A] :
            ( member(A,Y,A5)
           => ( aa(A,B,F3,Y) = aa(A,B,F3,Xb) ) )
       => ( the_elem(B,aa(set(A),set(B),image2(A,B,F3),A5)) = aa(A,B,F3,Xb) ) ) ) ).

% the_elem_image_unique
tff(fact_4429_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)))
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xb)
             => ? [X4: B] :
                  ( member(B,X4,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),aa(B,A,F3,X4)) ) ) ) ) ).

% le_SUP_iff
tff(fact_4430_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),Xb)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y4)
             => ? [X4: B] :
                  ( member(B,X4,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X4)),Y4) ) ) ) ) ).

% INF_le_iff
tff(fact_4431_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I6: set(A),C3: B,F3: fun(A,B)] :
          ( ( I6 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( member(A,I3,I6)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C3),aa(A,B,F3,I3)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),I6)) = C3 )
            <=> ! [X4: A] :
                  ( member(A,X4,I6)
                 => ( aa(A,B,F3,X4) = C3 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_4432_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),M4: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),M4) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),M4) ) ) ) ).

% cSUP_least
tff(fact_4433_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I6: set(A),F3: fun(A,B),C3: B] :
          ( ( I6 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( member(A,I3,I6)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),C3) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),I6)) = C3 )
            <=> ! [X4: A] :
                  ( member(A,X4,I6)
                 => ( aa(A,B,F3,X4) = C3 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4434_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),Mb: B,F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F3,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ).

% cINF_greatest
tff(fact_4435_card__image__le,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(A),nat,finite_card(A),A5)) ) ).

% card_image_le
tff(fact_4436_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mb(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H),I6) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G3,I6)),groups1962203154675924110t_prod(A,B,H,I6)) ) ) ) ).

% prod.distrib_triv'
tff(fact_4437_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G3),B5))) ) ) ) ).

% SUP_subset_mono
tff(fact_4438_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( ! [X3: A] :
                ( member(A,X3,B5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G3),B5))) ) ) ) ).

% INF_superset_mono
tff(fact_4439_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C3: A,A5: set(B)] :
          aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mc(A,fun(B,A),C3)),A5)) = $ite(A5 = bot_bot(set(B)),bot_bot(A),C3) ) ).

% SUP_constant
tff(fact_4440_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_4441_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S2: set(A),T2: set(B),G3: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),T2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G3),S2)),T2)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_me(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G3),H)),T2) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),H),S2) ) ) ) ) ) ).

% sum.group
tff(fact_4442_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I6: set(A),Xb: B,F3: fun(A,B)] :
          ( ( I6 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mf(B,fun(fun(A,B),fun(A,B)),Xb),F3)),I6)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Xb),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),I6))) ) ) ) ).

% INF_inf_const1
tff(fact_4443_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I6: set(A),F3: fun(A,B),Xb: B] :
          ( ( I6 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_mg(fun(A,B),fun(B,fun(A,B)),F3),Xb)),I6)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),I6))),Xb) ) ) ) ).

% INF_inf_const2
tff(fact_4444_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S2: set(A),T2: set(B),G3: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),T2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G3),S2)),T2)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mh(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G3),H)),T2) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),H),S2) ) ) ) ) ) ).

% prod.group
tff(fact_4445_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ).

% INF_le_SUP
tff(fact_4446_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),aa(set(A),set(B),image2(A,B,F3),A5))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B5)),aa(set(A),nat,finite_card(A),A5)) ) ) ).

% surj_card_le
tff(fact_4447_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,Xb: A,Yb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(set(A),set(A),image2(A,A,real_V8093663219630862766scaleR(A,C3)),set_or1337092689740270186AtMost(A,Xb,Yb)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C3),Xb),aa(A,A,real_V8093663219630862766scaleR(A,C3),Yb)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_4448_image__Suc__lessThan,axiom,
    ! [Nb: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),Nb) ).

% image_Suc_lessThan
tff(fact_4449_image__Suc__atMost,axiom,
    ! [Nb: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,Nb)) ).

% image_Suc_atMost
tff(fact_4450_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4451_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_4452_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_4453_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_4454_Fract__less__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A2,B2)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int)) ) ) ).

% Fract_less_zero_iff
tff(fact_4455_zero__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),fract(A2,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2) ) ) ).

% zero_less_Fract_iff
tff(fact_4456_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(A,$o)),I6),G3)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(A,$o)),I6),H)))
           => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mb(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H),I6) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G3,I6)),groups1962203154675924110t_prod(A,B,H,I6)) ) ) ) ) ).

% prod.distrib'
tff(fact_4457_one__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),fract(A2,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),A2) ) ) ).

% one_less_Fract_iff
tff(fact_4458_Fract__less__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A2,B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),B2) ) ) ).

% Fract_less_one_iff
tff(fact_4459_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P: fun(B,A),I6: set(B)] :
          groups1962203154675924110t_prod(B,A,P,I6) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_mi(fun(B,A),fun(set(B),fun(B,$o)),P),I6))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_mi(fun(B,A),fun(set(B),fun(B,$o)),P),I6))),one_one(A)) ) ).

% prod.G_def
tff(fact_4460_zero__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),fract(A2,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% zero_le_Fract_iff
tff(fact_4461_Fract__le__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),fract(A2,B2)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% Fract_le_zero_iff
tff(fact_4462_Fract__le__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),fract(A2,B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),B2) ) ) ).

% Fract_le_one_iff
tff(fact_4463_one__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),fract(A2,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2) ) ) ).

% one_le_Fract_iff
tff(fact_4464_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,Xb: A,Yb: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,Xb,Yb)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Yb)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Yb),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Xb)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4465_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,Xb: A,Yb: A] :
          aa(set(A),set(A),image2(A,A,aTP_Lamp_mj(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,Xb,Yb)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Yb),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C3))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4466_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C3: A,A2: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_mk(A,fun(A,fun(A,A)),Mb),C3)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C3))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4467_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C3: A,A2: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_ml(A,fun(A,fun(A,A)),Mb),C3)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C3))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4468_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C3: A,A2: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_mm(A,fun(A,fun(A,A)),Mb),C3)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Mb)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,Mb)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,Mb)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Mb)),C3))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4469_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C3: A,A2: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_mn(A,fun(A,fun(A,A)),Mb),C3)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Mb)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,Mb)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,Mb)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Mb)),C3))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4470_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S2: set(A),R: set(B),G3: fun(A,B),F3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),R)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G3),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_mo(fun(A,B),fun(fun(B,C),fun(A,C)),G3),F3)),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_mp(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S2),G3),F3)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_4471_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I2)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_4472_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: A,B5: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_mq(A,fun(nat,A),B5)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B5) ) ).

% INF_nat_binary
tff(fact_4473_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Xb),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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4474_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_mr(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_4475_ring__1__class_Oof__int__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_lb(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_4476_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_4477_case__prod__Pair,axiom,
    ! [B: $tType,A: $tType] : aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)) = id(product_prod(A,B)) ).

% case_prod_Pair
tff(fact_4478_id__funpow,axiom,
    ! [A: $tType,Nb: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),id(A)) = id(A) ).

% id_funpow
tff(fact_4479_remove1__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: A,F3: fun(A,B),Xs: list(A)] : remove1(A,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Xs)) = Xs ) ).

% remove1_insort_key
tff(fact_4480_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A5: set(product_prod(A,B)),F3: fun(A,fun(B,C))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),A5)
     => member(C,aa(B,C,aa(A,fun(B,C),F3,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),F3)),A5)) ) ).

% pair_imageI
tff(fact_4481_finite__UN,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A5)))
      <=> ! [X4: A] :
            ( member(A,X4,A5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,X4)) ) ) ) ).

% finite_UN
tff(fact_4482_push__bit__0__id,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4730199178511100633sh_bit(A,zero_zero(nat)) = id(A) ) ) ).

% push_bit_0_id
tff(fact_4483_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_4484_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_insort
tff(fact_4485_UN__constant,axiom,
    ! [B: $tType,A: $tType,C3: set(A),A5: 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_ms(set(A),fun(B,set(A)),C3)),A5)) = $ite(A5 = bot_bot(set(B)),bot_bot(set(A)),C3) ).

% UN_constant
tff(fact_4486_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [A4: A] :
            ( member(A,A4,A5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,A4)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A5))) ) ) ).

% finite_UN_I
tff(fact_4487_finite__INT,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B))] :
      ( ? [X: A] :
          ( member(A,X,I6)
          & aa(set(B),$o,finite_finite2(B),aa(A,set(B),A5,X)) )
     => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I6))) ) ).

% finite_INT
tff(fact_4488_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B5: fun(B,set(A)),C5: 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_mt(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B5)),C5)) = $ite(C5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C5)))) ).

% UN_simps(1)
tff(fact_4489_UN__singleton,axiom,
    ! [A: $tType,A5: 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_mu(A,set(A))),A5)) = A5 ).

% UN_singleton
tff(fact_4490_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Xb),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_4491_set__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),set(A),set2(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_4492_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,$o))),X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),S2),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S2)))) ) ).

% Inf_INT_eq2
tff(fact_4493_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,$o))),X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S2),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S2)))) ) ).

% Sup_SUP_eq2
tff(fact_4494_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S2: set(C),X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_mv(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S2)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),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_4495_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S2: set(C),X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_mv(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S2)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),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_4496_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o)))),S2)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),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_4497_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o)))),S2)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),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_4498_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Yb),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Yb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Xb),Xs)) ) ).

% insort_left_comm
tff(fact_4499_insort__key__left__comm,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xb: B,Yb: B,Xs: list(B)] :
          ( ( aa(B,A,F3,Xb) != aa(B,A,F3,Yb) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Yb),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Yb),Xs)) ) ) ) ).

% insort_key_left_comm
tff(fact_4500_insort__not__Nil,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),A2: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),A2),Xs) != nil(A) ) ).

% insort_not_Nil
tff(fact_4501_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3) = id(A) ).

% funpow_simps_right(1)
tff(fact_4502_None__notin__image__Some,axiom,
    ! [A: $tType,A5: set(A)] : ~ member(option(A),none(A),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A5)) ).

% None_notin_image_Some
tff(fact_4503_set__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_key
tff(fact_4504_distinct__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A,Xs: list(A)] :
          ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Xs))
        <=> ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
            & distinct(A,Xs) ) ) ) ).

% distinct_insort
tff(fact_4505_less__int__def,axiom,
    ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int_def
tff(fact_4506_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int_def
tff(fact_4507_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F4: fun(A,filter(B))] :
      ( ! [X7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B5)
         => ( aa(set(A),$o,finite_finite2(A),X7)
           => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),X7)) != bot_bot(filter(B)) ) ) )
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B5)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_4508_finite__int__iff__bounded__le,axiom,
    ! [S2: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S2)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S2)),aa(int,set(int),set_ord_atMost(int),K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_4509_finite__int__iff__bounded,axiom,
    ! [S2: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S2)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S2)),aa(int,set(int),set_ord_lessThan(int),K3)) ) ).

% finite_int_iff_bounded
tff(fact_4510_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A5: 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_mw(B,set(A))),A5)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_4511_UN__empty,axiom,
    ! [B: $tType,A: $tType,B5: 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),B5),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_4512_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A)),A5: 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),B5),A5)) )
    <=> ! [X4: B] :
          ( member(B,X4,A5)
         => ( aa(B,set(A),B5,X4) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_4513_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A)),A5: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A5)) = bot_bot(set(A)) )
    <=> ! [X4: B] :
          ( member(B,X4,A5)
         => ( aa(B,set(A),B5,X4) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_4514_nat__def,axiom,
    nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))) ).

% nat_def
tff(fact_4515_in__image__insert__iff,axiom,
    ! [A: $tType,B5: set(set(A)),Xb: A,A5: set(A)] :
      ( ! [C6: set(A)] :
          ( member(set(A),C6,B5)
         => ~ member(A,Xb,C6) )
     => ( member(set(A),A5,aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb)),B5))
      <=> ( member(A,Xb,A5)
          & member(set(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)),insert(A),Xb),bot_bot(set(A)))),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_4516_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A2,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_4517_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B5: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C5))) = $ite(C5 = bot_bot(set(B)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_mt(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B5)),C5))) ).

% UN_extend_simps(1)
tff(fact_4518_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A2,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_4519_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),C5: set(B),B5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),C5))),B5) = $ite(C5 = bot_bot(set(B)),B5,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_mx(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C5))) ).

% INT_extend_simps(1)
tff(fact_4520_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C5))) = $ite(C5 = bot_bot(set(B)),A5,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_my(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C5))) ).

% INT_extend_simps(2)
tff(fact_4521_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,A5: set(A),B11: set(set(A))] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)) = $ite(B11 = bot_bot(set(set(A))),A5,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)),A5)),B11))) ).

% Int_Inter_eq(1)
tff(fact_4522_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B11: set(set(A)),A5: 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)),A5) = $ite(B11 = bot_bot(set(set(A))),A5,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_mz(set(A),fun(set(A),set(A)),A5)),B11))) ).

% Int_Inter_eq(2)
tff(fact_4523_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X5: set(A),A5: set(product_prod(A,B)),Y5: set(B),P2: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ( X5 = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A5) )
     => ( ( Y5 = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),A5) )
       => ( ! [X3: A] :
              ( member(A,X3,X5)
             => ! [Xa3: B] :
                  ( member(B,Xa3,Y5)
                 => ( aa(B,$o,aa(A,fun(B,$o),P2,X3),Xa3)
                   => aa(B,$o,aa(A,fun(B,$o),Q,X3),Xa3) ) ) )
         => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P2)))
           => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_4524_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C5))) = $ite(C5 = bot_bot(set(B)),A5,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_na(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C5))) ).

% INT_extend_simps(4)
tff(fact_4525_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),K2: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_nb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M4),K2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2)))) ).

% UN_le_add_shift_strict
tff(fact_4526_UN__le__add__shift,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),K2: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_nb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M4),K2)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = 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,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2)))) ).

% UN_le_add_shift
tff(fact_4527_subset__subseqs,axiom,
    ! [A: $tType,X5: set(A),Xs: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(list(A),set(A),set2(A),Xs))
     => member(set(A),X5,aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% subset_subseqs
tff(fact_4528_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)),subseqs(A,Xs))) = pow2(A,aa(list(A),set(A),set2(A),Xs)) ).

% subseqs_powset
tff(fact_4529_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image2(int,int,aTP_Lamp_nc(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_4530_sum_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [I6: set(A),A5: fun(A,set(B)),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ! [X3: A] :
                ( member(A,X3,I6)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A5,X3)) )
           => ( ! [X3: A] :
                  ( member(A,X3,I6)
                 => ! [Xa3: A] :
                      ( member(A,Xa3,I6)
                     => ( ( X3 != Xa3 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,X3)),aa(A,set(B),A5,Xa3)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I6))) = 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_nd(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A5),G3)),I6) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_4531_prod_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [I6: set(A),A5: fun(A,set(B)),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ! [X3: A] :
                ( member(A,X3,I6)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A5,X3)) )
           => ( ! [X3: A] :
                  ( member(A,X3,I6)
                 => ! [Xa3: A] :
                      ( member(A,Xa3,I6)
                     => ( ( X3 != Xa3 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,X3)),aa(A,set(B),A5,Xa3)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I6))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ne(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A5),G3)),I6) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_4532_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I6)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I6)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_nf(fun(A,set(B)),fun(A,nat),A5)),I6)) ) ).

% card_UN_le
tff(fact_4533_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Xb),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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_4534_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),U)
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),aa(nat,set(nat),set_ord_lessThan(nat),aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_4535_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I6)
     => ( ! [X3: A] :
            ( member(A,X3,I6)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A5,X3)) )
       => ( ! [X3: A] :
              ( member(A,X3,I6)
             => ! [Xa3: A] :
                  ( member(A,Xa3,I6)
                 => ( ( X3 != Xa3 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,X3)),aa(A,set(B),A5,Xa3)) = 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),A5),I6))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_nf(fun(A,set(B)),fun(A,nat),A5)),I6) ) ) ) ) ).

% card_UN_disjoint
tff(fact_4536_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F3: fun(nat,set(A)),S2: set(A)] :
      ( ! [I3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F3,I3)),S2)
     => ( aa(set(A),$o,finite_finite2(A),S2)
       => ( ? [N7: nat] :
              ( ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N7)
                 => ! [M: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N7)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F3,M)),aa(nat,set(A),F3,N2)) ) ) )
              & ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
                 => ( aa(nat,set(A),F3,N7) = aa(nat,set(A),F3,N2) ) ) )
         => ( aa(nat,set(A),F3,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),F3),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_4537_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),remdups(A,concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),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_4538_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_na(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C5)) = $ite(C5 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C5)))) ).

% INT_simps(4)
tff(fact_4539_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [K3: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),K3)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_nh(real,fun(A,set(A)),E4)),K3))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_4540_top__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( top(A)
     => ! [Xb: B] : aa(B,A,top_top(fun(B,A)),Xb) = top_top(A) ) ).

% top_apply
tff(fact_4541_atMost__UNIV__triv,axiom,
    ! [A: $tType] : aa(set(A),set(set(A)),set_ord_atMost(set(A)),top_top(set(A))) = top_top(set(set(A))) ).

% atMost_UNIV_triv
tff(fact_4542_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_4543_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),top_top(set(option(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_4544_finite__Plus__UNIV__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(sum_sum(A,B)),$o,finite_finite2(sum_sum(A,B)),top_top(set(sum_sum(A,B))))
    <=> ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
        & aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_4545_inf__top_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),top_top(A)) = A2 ) ).

% inf_top.right_neutral
tff(fact_4546_inf__top_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A2: A,B2: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
        <=> ( ( A2 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_4547_inf__top_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),A2) = A2 ) ).

% inf_top.left_neutral
tff(fact_4548_inf__top_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = top_top(A) )
        <=> ( ( A2 = top_top(A) )
            & ( B2 = top_top(A) ) ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_4549_top__eq__inf__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xb: A,Yb: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) )
        <=> ( ( Xb = top_top(A) )
            & ( Yb = top_top(A) ) ) ) ) ).

% top_eq_inf_iff
tff(fact_4550_inf__eq__top__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = top_top(A) )
        <=> ( ( Xb = top_top(A) )
            & ( Yb = top_top(A) ) ) ) ) ).

% inf_eq_top_iff
tff(fact_4551_inf__top__right,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),top_top(A)) = Xb ) ).

% inf_top_right
tff(fact_4552_inf__top__left,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),Xb) = Xb ) ).

% inf_top_left
tff(fact_4553_range__mult,axiom,
    ! [A2: real] :
      aa(set(real),set(real),image2(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = $ite(A2 = zero_zero(real),aa(set(real),set(real),aa(real,fun(set(real),set(real)),insert(real),zero_zero(real)),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_4554_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),Xb) = top_top(A) ) ).

% max_top
tff(fact_4555_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_4556_dist__add__cancel2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [B2: A,A2: A,C3: A] : real_V557655796197034286t_dist(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),C3),A2)) = real_V557655796197034286t_dist(A,B2,C3) ) ).

% dist_add_cancel2
tff(fact_4557_dist__add__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C3: A] : real_V557655796197034286t_dist(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),C3)) = real_V557655796197034286t_dist(A,B2,C3) ) ).

% dist_add_cancel
tff(fact_4558_remdups__eq__nil__iff,axiom,
    ! [A: $tType,Xb: list(A)] :
      ( ( remdups(A,Xb) = nil(A) )
    <=> ( Xb = nil(A) ) ) ).

% remdups_eq_nil_iff
tff(fact_4559_remdups__eq__nil__right__iff,axiom,
    ! [A: $tType,Xb: list(A)] :
      ( ( nil(A) = remdups(A,Xb) )
    <=> ( Xb = nil(A) ) ) ).

% remdups_eq_nil_right_iff
tff(fact_4560_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_4561_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_4562_distinct__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : distinct(A,remdups(A,Xs)) ).

% distinct_remdups
tff(fact_4563_remdups__id__iff__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( remdups(A,Xs) = Xs )
    <=> distinct(A,Xs) ) ).

% remdups_id_iff_distinct
tff(fact_4564_Collect__const,axiom,
    ! [A: $tType,P2: $o] :
      aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ni($o,fun(A,$o),(P2))) = $ite((P2),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_4565_finite__Collect__not,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_nj(fun(A,$o),fun(A,$o),P2)))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_4566_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: $o] :
      aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_nk($o,fun(A,fun(B,$o)),(P2)))) = $ite((P2),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_4567_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_add
tff(fact_4568_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_4569_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A5) = top_top(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),top_top(A))
             => ? [Xa2: A] :
                  ( member(A,Xa2,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xa2) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_4570_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_4571_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_4572_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_4573_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_4574_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_4575_Diff__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_4576_surj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),Nb: nat] :
      ( ( aa(set(A),set(A),image2(A,A,F3),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)),Nb),F3)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_4577_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : real_V557655796197034286t_dist(A,zero_zero(A),Xb) = real_V7770717601297561774m_norm(A,Xb) ) ).

% dist_0_norm
tff(fact_4578_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Yb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Yb))
        <=> ( Xb != Yb ) ) ) ).

% zero_less_dist_iff
tff(fact_4579_finite__compl,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),uminus_uminus(set(A)),A5))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_compl
tff(fact_4580_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_remdups_leq
tff(fact_4581_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5)) = top_top(A) )
        <=> ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),top_top(A))
             => ? [Xa2: B] :
                  ( member(B,Xa2,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),aa(B,A,F3,Xa2)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_4582_range__constant,axiom,
    ! [B: $tType,A: $tType,Xb: A] : aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_kj(A,fun(B,A)),Xb)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ).

% range_constant
tff(fact_4583_ccINF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B)))) = top_top(A) ) ).

% ccINF_empty
tff(fact_4584_INT__constant,axiom,
    ! [B: $tType,A: $tType,C3: set(A),A5: 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_ms(set(A),fun(B,set(A)),C3)),A5)) = $ite(A5 = bot_bot(set(B)),top_top(set(A)),C3) ).

% INT_constant
tff(fact_4585_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),Xb)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_4586_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_my(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C5)) = $ite(C5 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C5)))) ).

% INT_simps(2)
tff(fact_4587_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),B5: set(A),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_mx(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C5)) = $ite(C5 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),C5))),B5)) ).

% INT_simps(1)
tff(fact_4588_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),B5: set(A),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_nl(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C5)) = $ite(C5 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),C5))),B5)) ).

% INT_simps(3)
tff(fact_4589_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))),insert(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_4590_surj__prod__encode,axiom,
    aa(set(product_prod(nat,nat)),set(nat),image2(product_prod(nat,nat),nat,nat_prod_encode),top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).

% surj_prod_encode
tff(fact_4591_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I6: set(A),F4: fun(A,filter(B))] :
      ( ! [I3: A] :
          ( member(A,I3,I6)
         => ! [J2: A] :
              ( member(A,J2,I6)
             => ? [X: A] :
                  ( member(A,X,I6)
                  & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I3)),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)) )
      <=> ? [X4: A] :
            ( member(A,X4,I6)
            & ( aa(A,filter(B),F4,X4) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_4592_remdups_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups(A,nil(A)) = nil(A) ).

% remdups.simps(1)
tff(fact_4593_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),top_top(A)) ) ).

% top_greatest
tff(fact_4594_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_4595_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_4596_infinite__UNIV__nat,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% infinite_UNIV_nat
tff(fact_4597_nat__not__finite,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% nat_not_finite
tff(fact_4598_infinite__UNIV__char__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% infinite_UNIV_char_0
tff(fact_4599_ex__new__if__finite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),A5)
       => ? [A4: A] : ~ member(A,A4,A5) ) ) ).

% ex_new_if_finite
tff(fact_4600_finite__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_UNIV
tff(fact_4601_Finite__Set_Ofinite__set,axiom,
    ! [A: $tType] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),top_top(set(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% Finite_Set.finite_set
tff(fact_4602_finite__Prod__UNIV,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(B),$o,finite_finite2(B),top_top(set(B)))
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B)))) ) ) ).

% finite_Prod_UNIV
tff(fact_4603_finite__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B))))
    <=> ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
        & aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_prod
tff(fact_4604_finite__fun__UNIVD2,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ).

% finite_fun_UNIVD2
tff(fact_4605_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_4606_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A2) ) ).

% top.extremum_strict
tff(fact_4607_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Yb: A,Xb: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Yb,Xb)),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Yb)),E3) ) ) ).

% dist_commute_lessI
tff(fact_4608_not__UNIV__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L2: A,H2: A] : top_top(set(A)) != set_or1337092689740270186AtMost(A,L2,H2) ) ).

% not_UNIV_eq_Icc
tff(fact_4609_atMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xb: A] :
          ( ( aa(A,set(A),set_ord_atMost(A),Xb) = top_top(set(A)) )
        <=> ( Xb = top_top(A) ) ) ) ).

% atMost_eq_UNIV_iff
tff(fact_4610_remdups__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : remdups(A,remdups(A,Xs)) = remdups(A,Xs) ).

% remdups_remdups
tff(fact_4611_not__UNIV__eq__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H2: A] : top_top(set(A)) != aa(A,set(A),set_ord_atMost(A),H2) ) ).

% not_UNIV_eq_Iic
tff(fact_4612_distinct__remdups__id,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups(A,Xs) = Xs ) ) ).

% distinct_remdups_id
tff(fact_4613_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_4614_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [Xb: A,Yb: A] :
          ( ( set_or1337092689740270186AtMost(A,Xb,Yb) = top_top(set(A)) )
        <=> ( ( Xb = bot_bot(A) )
            & ( Yb = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_4615_Inf__filter__not__bot,axiom,
    ! [A: $tType,B5: set(filter(A))] :
      ( ! [X7: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X7),B5)
         => ( aa(set(filter(A)),$o,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)),B5) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_4616_infinite__UNIV__int,axiom,
    ~ aa(set(int),$o,finite_finite2(int),top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_4617_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_4618_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_4619_infinite__UNIV__listI,axiom,
    ! [A: $tType] : ~ aa(set(list(A)),$o,finite_finite2(list(A)),top_top(set(list(A)))) ).

% infinite_UNIV_listI
tff(fact_4620_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : real_V7770717601297561774m_norm(A,Xb) = real_V557655796197034286t_dist(A,Xb,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_4621_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Yb: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Yb)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_4622_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Yb: A] :
          ( ( Xb != Yb )
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Yb)) ) ) ).

% dist_pos_lt
tff(fact_4623_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Yb: A,E1: real,X2: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Yb)),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Yb)),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_4624_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Z: A,Yb: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xb,Z)),real_V557655796197034286t_dist(A,Yb,Z))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Yb)),E3) ) ) ).

% dist_triangle_lt
tff(fact_4625_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_4626_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xb: A] : top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_4627_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H)) ) ).

% not_UNIV_le_Icc
tff(fact_4628_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),nat,finite_card(A),A5) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
       => ( A5 = top_top(set(A)) ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_4629_finite__range__Some,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_range_Some
tff(fact_4630_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H)) ) ).

% not_UNIV_le_Iic
tff(fact_4631_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_4632_notin__range__Some,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ~ member(option(A),Xb,aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A))))
    <=> ( Xb = none(A) ) ) ).

% notin_range_Some
tff(fact_4633_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_4634_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_4635_int__in__range__abs,axiom,
    ! [Nb: nat] : member(int,aa(nat,int,semiring_1_of_nat(int),Nb),aa(set(int),set(int),image2(int,int,abs_abs(int)),top_top(set(int)))) ).

% int_in_range_abs
tff(fact_4636_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_4637_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G3: fun(B,A),F3: fun(A,C)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,G3),top_top(set(B))))
     => aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_nm(fun(B,A),fun(fun(A,C),fun(B,C)),G3),F3)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_4638_remove1__remdups,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( remove1(A,Xb,remdups(A,Xs)) = remdups(A,remove1(A,Xb,Xs)) ) ) ).

% remove1_remdups
tff(fact_4639_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A2: A,Xb: B] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) )
     => ( aa(B,A,F3,Xb) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4640_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_4641_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C3: A,A5: set(B)] :
          aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_mc(A,fun(B,A),C3)),A5)) = $ite(A5 = bot_bot(set(B)),top_top(A),C3) ) ).

% INF_constant
tff(fact_4642_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_4643_INT__empty,axiom,
    ! [B: $tType,A: $tType,B5: 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),B5),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_4644_UN__UN__finite__eq,axiom,
    ! [A: $tType,A5: 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_nn(fun(nat,set(A)),fun(nat,set(A)),A5)),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),A5),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_4645_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M8: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M2),aa(nat,A,X5,N))),E4) ) ) ) ) ) ).

% Cauchy_def
tff(fact_4646_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [N6: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S,N),aa(nat,A,S,N6))),E4) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_4647_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ? [M7: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M3)
               => ! [N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N4)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M3),aa(nat,A,X5,N4))),E3) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_4648_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M),aa(nat,A,X5,N2))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% metric_CauchyI
tff(fact_4649_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_4650_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_4651_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Yb: A,E3: real,X2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Yb)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Yb)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),E3) ) ) ) ).

% dist_triangle_half_l
tff(fact_4652_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Yb: A,X1: A,E3: real,X2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Yb,X1)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Yb,X2)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),E3) ) ) ) ).

% dist_triangle_half_r
tff(fact_4653_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,X2: A,E3: real,X32: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),bit1(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,X32)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),bit1(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),bit1(one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X42)),E3) ) ) ) ) ).

% dist_triangle_third
tff(fact_4654_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M),aa(nat,A,X5,N2))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI'
tff(fact_4655_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F3: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F3)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M8: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F3,M2),aa(nat,A,F3,N))),E4) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_4656_UN__finite__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),C5: set(A)] :
      ( ! [N2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),C5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),top_top(set(nat))))),C5) ) ).

% UN_finite_subset
tff(fact_4657_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_4658_UN__finite2__eq,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B5: fun(nat,set(A)),K2: nat] :
      ( ! [N2: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),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),B5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K2))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),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),B5),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_4659_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),C5: set(B),B5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),C5))),B5) = $ite(C5 = bot_bot(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),B5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_nl(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C5))) ).

% INT_extend_simps(3)
tff(fact_4660_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F3),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_4661_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [J3: nat] :
            ? [M8: nat] :
            ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
             => ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M2),aa(nat,A,X5,N))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_4662_UN__finite2__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B5: fun(nat,set(A)),K2: nat] :
      ( ! [N2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K2)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A5),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),B5),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_4663_range__mod,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_no(nat,fun(nat,nat),Nb)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_4664_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_4665_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_4666_cclfp__def,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F3: fun(A,A)] : order_532582986084564980_cclfp(A,F3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_np(fun(A,A),fun(nat,A),F3)),top_top(set(nat)))) ) ).

% cclfp_def
tff(fact_4667_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa(real,real,root(Nb),Xb) = $ite(Nb = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_nq(nat,fun(real,real),Nb),Xb)) ).

% root_def
tff(fact_4668_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_4669_less__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F8: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),F4),F8)
    <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F8)
        & ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F8),F4) ) ) ).

% less_filter_def
tff(fact_4670_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_4671_mlex__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F3,R) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_nr(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F3),R))) ).

% mlex_eq
tff(fact_4672_these__insert__Some,axiom,
    ! [A: $tType,Xb: A,A5: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),aa(A,option(A),some(A),Xb)),A5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),these(A,A5)) ).

% these_insert_Some
tff(fact_4673_Id__on__def,axiom,
    ! [A: $tType,A5: set(A)] : id_on(A,A5) = 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_ns(A,set(product_prod(A,A)))),A5)) ).

% Id_on_def
tff(fact_4674_top2I,axiom,
    ! [A: $tType,B: $tType,Xb: A,Yb: B] : aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),Xb),Yb) ).

% top2I
tff(fact_4675_Id__onI,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( member(A,A2,A5)
     => 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,A5)) ) ).

% Id_onI
tff(fact_4676_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_4677_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_4678_these__image__Some__eq,axiom,
    ! [A: $tType,A5: set(A)] : these(A,aa(set(A),set(option(A)),image2(A,option(A),some(A)),A5)) = A5 ).

% these_image_Some_eq
tff(fact_4679_these__insert__None,axiom,
    ! [A: $tType,A5: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),A5)) = these(A,A5) ).

% these_insert_None
tff(fact_4680_Id__on__iff,axiom,
    ! [A: $tType,Xb: A,Yb: A,A5: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),id_on(A,A5))
    <=> ( ( Xb = Yb )
        & member(A,Xb,A5) ) ) ).

% Id_on_iff
tff(fact_4681_Id__on__eqI,axiom,
    ! [A: $tType,A2: A,B2: A,A5: set(A)] :
      ( ( A2 = B2 )
     => ( member(A,A2,A5)
       => 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,A5)) ) ) ).

% Id_on_eqI
tff(fact_4682_Id__onE,axiom,
    ! [A: $tType,C3: product_prod(A,A),A5: set(A)] :
      ( member(product_prod(A,A),C3,id_on(A,A5))
     => ~ ! [X3: A] :
            ( member(A,X3,A5)
           => ( C3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ) ) ).

% Id_onE
tff(fact_4683_in__these__eq,axiom,
    ! [A: $tType,Xb: A,A5: set(option(A))] :
      ( member(A,Xb,these(A,A5))
    <=> member(option(A),aa(A,option(A),some(A),Xb),A5) ) ).

% in_these_eq
tff(fact_4684_these__not__empty__eq,axiom,
    ! [A: $tType,B5: set(option(A))] :
      ( ( these(A,B5) != bot_bot(set(A)) )
    <=> ( ( B5 != bot_bot(set(option(A))) )
        & ( B5 != aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_4685_these__empty__eq,axiom,
    ! [A: $tType,B5: set(option(A))] :
      ( ( these(A,B5) = bot_bot(set(A)) )
    <=> ( ( B5 = bot_bot(set(option(A))) )
        | ( B5 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_4686_Some__image__these__eq,axiom,
    ! [A: $tType,A5: set(option(A))] : aa(set(A),set(option(A)),image2(A,option(A),some(A)),these(A,A5)) = aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_nt(set(option(A)),fun(option(A),$o),A5)) ).

% Some_image_these_eq
tff(fact_4687_mlex__leq,axiom,
    ! [A: $tType,F3: fun(A,nat),Xb: A,Yb: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Xb)),aa(A,nat,F3,Yb))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),mlex_prod(A,F3,R)) ) ) ).

% mlex_leq
tff(fact_4688_mlex__iff,axiom,
    ! [A: $tType,Xb: A,Yb: A,F3: fun(A,nat),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),mlex_prod(A,F3,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xb)),aa(A,nat,F3,Yb))
        | ( ( aa(A,nat,F3,Xb) = aa(A,nat,F3,Yb) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R) ) ) ) ).

% mlex_iff
tff(fact_4689_mlex__less,axiom,
    ! [A: $tType,F3: fun(A,nat),Xb: A,Yb: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xb)),aa(A,nat,F3,Yb))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),mlex_prod(A,F3,R)) ) ).

% mlex_less
tff(fact_4690_in__finite__psubset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A5),B5),finite_psubset(A))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
        & aa(set(A),$o,finite_finite2(A),B5) ) ) ).

% in_finite_psubset
tff(fact_4691_in__measure,axiom,
    ! [A: $tType,Xb: A,Yb: A,F3: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),measure(A,F3))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xb)),aa(A,nat,F3,Yb)) ) ).

% in_measure
tff(fact_4692_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
         => has_field_derivative(real,root(Nb),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Nb))),aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_4693_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_4694_at__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [Xb: A,S2: set(A)] : topolo174197925503356063within(A,Xb,S2) = bot_bot(filter(A)) ) ).

% at_discrete
tff(fact_4695_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Yb: A,Z: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,F3,Yb,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S2)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_nu(fun(A,A),fun(A,fun(A,A)),F3),Z),Yb,topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_at_within_shift
tff(fact_4696_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,Xb: A,S: set(A),G3: fun(A,A),Db: A] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nv(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G3,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F3,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult
tff(fact_4697_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A,S: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nv(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),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,F3,Xb)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G3,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult'
tff(fact_4698_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A,S: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nw(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),D5),E5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_add
tff(fact_4699_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K2: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_nx(A,fun(A,A),K2),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_4700_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F9: A,F4: filter(A),G3: fun(A,A),G6: A] :
          ( has_field_derivative(A,F3,F9,F4)
         => ( has_field_derivative(A,G3,G6,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nw(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),F9),G6),F4) ) ) ) ).

% field_differentiable_add
tff(fact_4701_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F9: A,A2: A,S2: set(A),D3: real,G3: fun(A,A)] :
          ( has_field_derivative(A,F3,F9,topolo174197925503356063within(A,A2,S2))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
           => ( member(A,A2,S2)
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D3)
                     => ( aa(A,A,F3,X3) = aa(A,A,G3,X3) ) ) )
               => has_field_derivative(A,G3,F9,topolo174197925503356063within(A,A2,S2)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_4702_has__real__derivative__neg__dec__left,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xb)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_4703_has__real__derivative__pos__inc__left,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3))),aa(real,real,F3,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_4704_has__real__derivative__pos__inc__right,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xb)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_4705_has__real__derivative__neg__dec__right,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))),aa(real,real,F3,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_4706_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,Xb: real,Yb: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( member(real,Xb,set_or5935395276787703475ssThan(real,A2,B2))
       => ( member(real,Yb,set_or5935395276787703475ssThan(real,A2,B2))
         => ( ! [X3: real] :
                ( member(real,X3,set_or5935395276787703475ssThan(real,A2,B2))
               => has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) )
           => ( aa(real,real,F3,Xb) = aa(real,real,F3,Yb) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_4707_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Yb: A,Xb: A,Z: A] :
          ( has_field_derivative(A,F3,Yb,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_ny(fun(A,A),fun(A,fun(A,A)),F3),Z),Yb,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_4708_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,B2)),aa(real,real,F3,A2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_4709_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y3) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,A2)),aa(real,real,F3,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_4710_DERIV__pos__inc__right,axiom,
    ! [F3: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xb)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_4711_DERIV__neg__dec__right,axiom,
    ! [F3: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))),aa(real,real,F3,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_4712_DERIV__neg__dec__left,axiom,
    ! [F3: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xb)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_4713_DERIV__pos__inc__left,axiom,
    ! [F3: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3))),aa(real,real,F3,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_4714_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A,S: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G3,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nz(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G3,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F3,Xb)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,Xb)),aa(A,A,G3,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_4715_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F3,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_oa(fun(A,A),fun(A,A),F3),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,F3,Xb))),D5)),aa(A,A,inverse_inverse(A),aa(A,A,F3,Xb)))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_4716_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_4717_trivial__limit__at__left__real,axiom,
    ! [A: $tType] :
      ( ( dense_order(A)
        & no_bot(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A] : topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_lessThan(A),Xb)) != bot_bot(filter(A)) ) ).

% trivial_limit_at_left_real
tff(fact_4718_MVT2,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),F9: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
             => has_field_derivative(real,F3,aa(real,real,F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B2)
            & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,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,Z2)) ) ) ) ) ).

% MVT2
tff(fact_4719_DERIV__local__const,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,D3: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y))),D3)
             => ( aa(real,real,F3,Xb) = aa(real,real,F3,Y) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_4720_DERIV__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_4721_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K2: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_ob(A,fun(A,A),K2),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xaa),K2))),topolo174197925503356063within(A,Xaa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_4722_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_oc(fun(A,A),fun(nat,fun(A,A)),F3),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,power_power(A,aa(A,A,F3,Xb)),Nb))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power_Suc
tff(fact_4723_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,S: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_inverse
tff(fact_4724_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_od(fun(A,A),fun(nat,fun(A,A)),F3),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,power_power(A,aa(A,A,F3,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power
tff(fact_4725_DERIV__local__max,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,D3: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y))),D3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F3,Y)),aa(real,real,F3,Xb)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_4726_DERIV__local__min,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,D3: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y))),D3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F3,Xb)),aa(real,real,F3,Y)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_4727_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),divide_divide(real,one_one(real),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_4728_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_oe(nat,fun(real,real),Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,power_power(real,Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xb,S)) ).

% DERIV_pow
tff(fact_4729_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo174197925503356063within(A,Xb,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,Xb,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_4730_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( topolo174197925503356063within(A,B2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_4731_trivial__limit__at__left__bot,axiom,
    ! [A: $tType] :
      ( ( order_bot(A)
        & topolo1944317154257567458pology(A) )
     => ( topolo174197925503356063within(A,bot_bot(A),aa(A,set(A),set_ord_lessThan(A),bot_bot(A))) = bot_bot(filter(A)) ) ) ).

% trivial_limit_at_left_bot
tff(fact_4732_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,Xb: A,S: set(A),G3: fun(A,A),E3: A] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G3,E3,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G3,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nz(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),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),D3),aa(A,A,G3,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),E3),aa(A,A,F3,Xb))),aa(nat,A,power_power(A,aa(A,A,G3,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_4733_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F3,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_oa(fun(A,A),fun(A,A),F3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,aa(A,A,F3,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_4734_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C3: fun(nat,A),F3: fun(A,A),F9: A,Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),C3),Z2),aa(A,A,F3,Z2)) )
         => ( has_field_derivative(A,F3,F9,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_of(fun(nat,A),fun(A,fun(nat,A)),C3),Z),F9) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_4735_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_og(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_4736_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C3: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),C3),Z2)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
           => has_field_derivative(A,aTP_Lamp_oh(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_of(fun(nat,A),fun(A,fun(nat,A)),C3),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_4737_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => has_field_derivative(A,aTP_Lamp_oh(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_of(fun(nat,A),fun(A,fun(nat,A)),C3),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_4738_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_of(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_oi(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
               => has_field_derivative(A,aTP_Lamp_oh(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_of(fun(nat,A),fun(A,fun(nat,A)),C3),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_4739_DERIV__log,axiom,
    ! [Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,log(B2),divide_divide(real,one_one(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_4740_DERIV__fun__powr,axiom,
    ! [G3: fun(real,real),Mb: real,Xb: real,R2: real] :
      ( has_field_derivative(real,G3,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G3,Xb))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_oj(fun(real,real),fun(real,fun(real,real)),G3),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,G3,Xb),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),Mb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_4741_DERIV__powr,axiom,
    ! [G3: fun(real,real),Mb: real,Xb: real,F3: fun(real,real),R2: real] :
      ( has_field_derivative(real,G3,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G3,Xb))
       => ( has_field_derivative(real,F3,R2,topolo174197925503356063within(real,Xb,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_ok(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F3),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G3,Xb),aa(real,real,F3,Xb))),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,G3,Xb)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),Mb),aa(real,real,F3,Xb)),aa(real,real,G3,Xb)))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_4742_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_4743_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A5: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => 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,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A5)) ) ).

% arcosh_real_has_field_derivative
tff(fact_4744_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,sqrt,divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_4745_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_4746_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Xb: A,Db: A,S: set(A)] :
          ( ( cosh(A,aa(A,A,G3,Xb)) != zero_zero(A) )
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aTP_Lamp_ol(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,power_power(A,aa(A,A,tanh(A),aa(A,A,G3,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_4747_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D5: real] :
      ( ( Xb != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( D5 = divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
           => ( D5 = divide_divide(real,aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_4748_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A5: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => has_field_derivative(real,artanh(real),divide_divide(real,one_one(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xb,A5)) ) ).

% artanh_real_has_field_derivative
tff(fact_4749_DERIV__power__series_H,axiom,
    ! [R: real,F3: fun(nat,real),X0: real] :
      ( ! [X3: real] :
          ( member(real,X3,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_om(fun(nat,real),fun(real,fun(nat,real)),F3),X3)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => has_field_derivative(real,aTP_Lamp_oo(fun(nat,real),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),aTP_Lamp_om(fun(nat,real),fun(real,fun(nat,real)),F3),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_4750_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => has_field_derivative(real,root(Nb),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_4751_DERIV__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_4752_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Xb: real,Nb: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F3,Xb) = 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_op(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_4753_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Xb: real,Nb: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
        & ! [M: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ? [T5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xb))
          & ( aa(real,real,F3,Xb) = 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_op(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_4754_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( ( Xb != zero_zero(real) )
       => has_field_derivative(real,root(Nb),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_4755_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
         => ( ! [M: nat,T5: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
           => ? [T5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),H)
                & ( aa(real,real,F3,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_oq(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_4756_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F3: fun(real,real),Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M: nat,T5: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ? [T5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),H)
              & ( aa(real,real,F3,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_oq(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_4757_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),zero_zero(real))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
         => ( ! [M: nat,T5: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H),T5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
           => ? [T5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),zero_zero(real))
                & ( aa(real,real,F3,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_oq(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_4758_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( Xb != zero_zero(real) )
         => ( ! [M: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
           => ? [T5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T5))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xb))
                & ( aa(real,real,F3,Xb) = 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_op(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_4759_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M: nat,T5: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xb)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F3,Xb) = 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_op(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_4760_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M: nat,T5: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),B2)
             => ? [T5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),T5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),C3)
                  & ( aa(real,real,F3,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_or(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C3)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_4761_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M: nat,T5: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),B2)
             => ? [T5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),T5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),B2)
                  & ( aa(real,real,F3,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_or(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C3)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C3)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_4762_Taylor,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M: nat,T5: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),B2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xb)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),B2)
                 => ( ( Xb != C3 )
                   => ? [T5: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),C3),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T5)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),C3) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),T5)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),Xb) ) )
                        & ( aa(real,real,F3,Xb) = 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_os(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C3),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Nb),T5),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),C3)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_4763_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_ot(set(A),fun(set(A),$o)))) ).

% finite_psubset_def
tff(fact_4764_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun(nat,fun(real,real)),K2: nat,B5: real] :
      ( ! [M: nat,T5: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
     => ( ( Nb = aa(nat,nat,suc,K2) )
       => ! [M3: nat,T6: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Nb)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T6)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),H) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_ov(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Nb),Diff),B5),M3),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T6)),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_ow(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M3),T6)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M3))))),aa(real,real,aa(real,fun(real,real),times_times(real),B5),divide_divide(real,aa(nat,real,power_power(real,T6),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M3))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M3))))))),topolo174197925503356063within(real,T6,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_4765_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_ox(real,real),suminf(real,aTP_Lamp_oy(real,fun(nat,real),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_4766_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D5: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( Xb != zero_zero(real) )
       => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
             => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
               => ( D5 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
               => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Nb),D5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_4767_DERIV__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,power_power(real,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_4768_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G6: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G3,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G3,Xb)),one_one(real))
           => ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_oz(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pa(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xb),G6),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_4769_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G6: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xb))
         => ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_pb(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xb),G6),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_4770_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,real),F9: fun(A,real),Xb: A,S: set(A),G3: fun(A,B),G6: fun(A,B)] :
          ( has_derivative(A,real,F3,F9,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G3,G6,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pd(fun(A,real),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pe(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F9),Xb),G3),G6),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_4771_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C3: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_pf(B,fun(A,B),C3),aTP_Lamp_pg(A,B),F4) ) ).

% has_derivative_const
tff(fact_4772_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),F4: filter(A),G3: fun(A,B),G6: fun(A,B)] :
          ( has_derivative(A,B,F3,F9,F4)
         => ( has_derivative(A,B,G3,G6,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ph(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aTP_Lamp_ph(fun(A,B),fun(fun(A,B),fun(A,B)),F9),G6),F4) ) ) ) ).

% has_derivative_add
tff(fact_4773_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F4: fun(A,B),Xb: A] :
          ( has_derivative(A,B,aTP_Lamp_pg(A,B),F4,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => ! [X: A] : aa(A,B,F4,X) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_4774_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),Xb: A,S: set(A),G3: fun(A,B),G6: fun(A,B)] :
          ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G3,G6,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pi(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pj(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F9),Xb),G3),G6),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_mult
tff(fact_4775_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),Xb: A,S: set(A),D3: real,G3: fun(A,B)] :
          ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
           => ( member(A,Xb,S)
             => ( ! [X8: A] :
                    ( member(A,X8,S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xb)),D3)
                     => ( aa(A,B,F3,X8) = aa(A,B,G3,X8) ) ) )
               => has_derivative(A,B,G3,F9,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_4776_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),Xb: A,S2: set(A),G3: fun(A,B),G6: fun(A,B)] :
          ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,S2))
         => ( has_derivative(A,B,G3,G6,topolo174197925503356063within(A,Xb,S2))
           => ( ( aa(A,B,G3,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pk(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pl(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F9),Xb),G3),G6),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_4777_arccos__less__arccos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Yb),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Yb)),aa(real,real,arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_4778_arccos__less__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Yb)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Xb)),aa(real,real,arccos,Yb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_4779_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(B,A),Xb: B,F9: fun(B,A),S2: set(B)] :
          ( ( aa(B,A,F3,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F3,F9,topolo174197925503356063within(B,Xb,S2))
           => has_derivative(B,A,aTP_Lamp_pm(fun(B,A),fun(B,A),F3),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_pn(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F3),Xb),F9),topolo174197925503356063within(B,Xb,S2)) ) ) ) ).

% has_derivative_inverse
tff(fact_4780_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,S2: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_po(A,fun(A,A),Xb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_inverse'
tff(fact_4781_arccos__lt__bounded,axiom,
    ! [Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Yb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Yb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arccos,Yb))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Yb)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_4782_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G6: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xb))
         => ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_pp(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xb),G6),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_ln
tff(fact_4783_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xb)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_4784_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G6: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G3,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G3,Xb)),one_one(real))
           => ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_pr(fun(A,real),fun(A,real),G3),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))),G3),Xb),G6),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_4785_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),Xb: A,S2: set(A),G3: fun(A,B),G6: fun(A,B)] :
          ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,S2))
         => ( has_derivative(A,B,G3,G6,topolo174197925503356063within(A,Xb,S2))
           => ( ( aa(A,B,G3,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pt(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pu(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F9),Xb),G3),G6),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% has_derivative_divide
tff(fact_4786_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I6: set(A),F3: fun(A,fun(B,C)),F9: fun(A,fun(B,C)),Xb: B,S2: set(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I6)
             => has_derivative(B,C,aa(A,fun(B,C),F3,I3),aa(A,fun(B,C),F9,I3),topolo174197925503356063within(B,Xb,S2)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_pw(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I6),F3),aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_py(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I6),F3),F9),Xb),topolo174197925503356063within(B,Xb,S2)) ) ) ).

% has_derivative_prod
tff(fact_4787_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G6: fun(A,real),Xb: A,X5: set(A),F3: fun(A,real),F9: fun(A,real)] :
          ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xb,X5))
         => ( has_derivative(A,real,F3,F9,topolo174197925503356063within(A,Xb,X5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xb))
             => ( member(A,Xb,X5)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pz(fun(A,real),fun(fun(A,real),fun(A,real)),G3),F3),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))))),G3),G6),Xb),F3),F9),topolo174197925503356063within(A,Xb,X5)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_4788_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_oi(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qc(fun(nat,A),fun(A,fun(A,A)),C3),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_4789_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_qe(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_4790_Gcd__eq__Max,axiom,
    ! [M4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M4)
     => ( ( M4 != bot_bot(set(nat)) )
       => ( ~ 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_qf(nat,set(nat))),M4))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_4791_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = Xb ) ).

% Max_singleton
tff(fact_4792_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R2: set(product_prod(A,A)),S: set(product_prod(B,B))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)),lex_prod(A,B,R2,S))
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A3),R2)
        | ( ( A2 = A3 )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B3),S) ) ) ) ).

% in_lex_prod
tff(fact_4793_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ap(nat,fun(nat,$o),Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_4794_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xb)
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xb) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_4795_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xb)
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xb) ) ) ) ) ) ).

% Max_less_iff
tff(fact_4796_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(B,A),L: A,F4: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qg(A,fun(fun(B,A),fun(B,A)),C3),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C3)),F4)
          <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_4797_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(B,A),L: A,F4: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qh(A,fun(fun(B,A),fun(B,A)),C3),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),L)),F4)
          <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_4798_power__tendsto__0__iff,axiom,
    ! [A: $tType,Nb: nat,F3: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qi(nat,fun(fun(A,real),fun(A,real)),Nb),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_4799_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A5: set(A),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_qj(B,fun(A,B),C3)),A5)) = C3 ) ) ) ) ).

% Max_const
tff(fact_4800_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ) ).

% Max_insert
tff(fact_4801_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K2: A,A2: B] :
          ( ( K2 != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_qk(A,fun(B,A),K2),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_4802_tendsto__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F4: filter(A),F3: fun(A,B),A2: B,B2: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
           => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,B2),F4)
             => ( A2 = B2 ) ) ) ) ) ).

% tendsto_unique
tff(fact_4803_tendsto__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F3: fun(A,B),A2: B] : filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),bot_bot(filter(A))) ) ).

% tendsto_bot
tff(fact_4804_nhds__neq__bot,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo7230453075368039082e_nhds(A,A2) != bot_bot(filter(A)) ) ).

% nhds_neq_bot
tff(fact_4805_tendsto__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F4: filter(A),A2: B,B2: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,aTP_Lamp_ql(B,fun(A,B),A2),topolo7230453075368039082e_nhds(B,B2),F4)
          <=> ( A2 = B2 ) ) ) ) ).

% tendsto_const_iff
tff(fact_4806_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_qm(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F4) ) ) ) ).

% tendsto_cot
tff(fact_4807_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( cosh(B,A2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_qn(fun(A,B),fun(A,B),F3),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A2)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_4808_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_qo(fun(A,B),fun(A,B),F3),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A2)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_4809_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),G3: fun(A,B),F4: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qp(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
          <=> filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_4810_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_4811_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qp(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform2
tff(fact_4812_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G3: fun(A,B),A2: B,F4: filter(A),F3: fun(A,B)] :
          ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qr(fun(A,B),fun(fun(A,B),fun(A,B)),G3),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform
tff(fact_4813_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_4814_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qq(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_4815_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B),B2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,B2),F4)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qs(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,divide_divide(B,A2,B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_4816_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qt(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_4817_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B),B2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qu(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),F4) ) ) ) ).

% tendsto_add
tff(fact_4818_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C3: B,F3: fun(A,B),D3: B,F4: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qv(B,fun(fun(A,B),fun(A,B)),C3),F3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C3),D3)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,D3),F4) ) ) ).

% tendsto_add_const_iff
tff(fact_4819_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qu(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_4820_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qw(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_4821_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qx(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_4822_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qy(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_4823_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_qz(fun(A,B),fun(A,B),F3),topolo7230453075368039082e_nhds(B,sgn_sgn(B,L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_4824_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_ra(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_4825_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_ra(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_4826_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_ra(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_4827_tendsto__max,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),Xb: B,Net: filter(A),Y5: fun(A,B),Yb: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,Xb),Net)
         => ( filterlim(A,B,Y5,topolo7230453075368039082e_nhds(B,Yb),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rb(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y5),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),ord_max(B),Xb),Yb)),Net) ) ) ) ).

% tendsto_max
tff(fact_4828_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I6: set(A),F3: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I6)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_rc(fun(B,fun(A,C)),fun(A,fun(B,C)),F3),I3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_rd(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I6),F3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_4829_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_re(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F4) ) ) ) ).

% tendsto_tan
tff(fact_4830_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,C),B2: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,B2),F4)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_rf(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3),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_4831_tendsto__arcosh,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_rg(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ).

% tendsto_arcosh
tff(fact_4832_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L: B,Xb: A,S2: set(A),D3: real,G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xb,S2))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
           => ( ! [X8: A] :
                  ( member(A,X8,S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X8,Xb))
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xb)),D3)
                     => ( aa(A,B,F3,X8) = aa(A,B,G3,X8) ) ) ) )
             => filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% Lim_transform_within
tff(fact_4833_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_rh(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_4834_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_rh(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_4835_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_rh(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_4836_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F3: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_ri(fun(A,B),fun(nat,fun(A,B)),F3),Nb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_4837_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,K2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_rj(fun(A,B),fun(A,fun(A,B)),F3),K2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),K2),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_4838_tendsto__log,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
         => ( ( A2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rk(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_4839_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ).

% Max.coboundedI
tff(fact_4840_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ! [X3: A] :
                  ( member(A,X3,A5)
                 => ? [Xa: A] :
                      ( member(A,Xa,B5)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,B5)
                   => ? [Xa: A] :
                        ( member(A,Xa,A5)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,lattic643756798349783984er_Max(A),B5) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_4841_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [Y: A] :
                ( member(A,Y,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) )
           => ( member(A,Xb,A5)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = Xb ) ) ) ) ) ).

% Max_eqI
tff(fact_4842_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ).

% Max_ge
tff(fact_4843_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A5),A5) ) ) ) ).

% Max_in
tff(fact_4844_tendsto__artanh,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),one_one(real))
         => filterlim(A,real,aTP_Lamp_rl(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_4845_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(A),A,lattic643756798349783984er_Max(A),A5) ) ) ) ) ).

% Max.in_idem
tff(fact_4846_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X4),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_4847_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [S3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
                & ! [X: A] :
                    ( ( ( X != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),S3) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X),L5)),R2) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_4848_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F3: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S8: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S8)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),S8) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X3),L5)),R3) ) ) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_4849_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G3: fun(A,B),L: B,A2: A,R: real,F3: fun(A,B)] :
          ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ( ! [X3: A] :
                  ( ( X3 != A2 )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),R)
                   => ( aa(A,B,F3,X3) = aa(A,B,G3,X3) ) ) )
             => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_4850_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),B2: B,A2: A,G3: fun(B,C),C3: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G3,topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D4) )
                     => ( aa(A,B,F3,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rm(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_4851_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,F3: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( filterlim(B,C,F3,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_rn(B,fun(fun(B,C),fun(B,C)),A2),F3),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_4852_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F3: fun(A,B),G3: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),R)
                 => ( aa(A,B,F3,X3) = aa(A,B,G3,X3) ) ) )
           => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_4853_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,X4)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_4854_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F3: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S8: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S8)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),S8) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,X3)),L5))),R3) ) ) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_4855_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [S3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
                & ! [X: A] :
                    ( ( ( X != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),S3) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,X)),L5))),R2) ) ) ) ) ) ).

% LIM_D
tff(fact_4856_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,A),A2: A,D5: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ro(fun(A,A),fun(A,fun(A,A)),F3),A2),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rp(fun(A,A),fun(A,fun(A,A)),F3),A2),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_4857_LIM__fun__less__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X: real] :
                ( ( ( X != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C3),X))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,X)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_4858_LIM__fun__not__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X: real] :
                ( ( ( X != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C3),X))),R3) )
               => ( aa(real,real,F3,X) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_4859_LIM__fun__gt__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X: real] :
                ( ( ( X != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C3),X))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F3,X)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_4860_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),B2: B,A2: A,G3: fun(B,C),C3: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G3,topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),D4) )
                     => ( aa(A,B,F3,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rq(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_4861_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xb) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xb) ) ) ) ) ).

% Max.boundedI
tff(fact_4862_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xb)
             => ! [A9: A] :
                  ( member(A,A9,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),Xb) ) ) ) ) ) ).

% Max.boundedE
tff(fact_4863_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798349783984er_Max(A),A5) )
            <=> ( member(A,Mb,A5)
                & ! [X4: A] :
                    ( member(A,X4,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Mb) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4864_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5))
            <=> ? [X4: A] :
                  ( member(A,X4,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X4) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_4865_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = Mb )
            <=> ( member(A,Mb,A5)
                & ! [X4: A] :
                    ( member(A,X4,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Mb) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4866_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5))
            <=> ? [X4: A] :
                  ( member(A,X4,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X4) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_4867_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [B4: A] :
                ( member(A,B4,A5)
               => aa(A,$o,aa(A,fun(A,$o),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)),insert(A),A2),A5)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_4868_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ) ) ) ).

% Max_Sup
tff(fact_4869_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = aa(set(A),A,lattic643756798349783984er_Max(A),X5) ) ) ) ) ).

% cSup_eq_Max
tff(fact_4870_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rr(fun(A,A),fun(A,fun(A,A)),F3),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_4871_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rr(fun(A,A),fun(A,fun(A,A)),F3),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_4872_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_rs(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_4873_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K2: real,F3: fun(A,B),K5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
         => ( ! [H4: A] :
                ( ( H4 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H4)),K2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,H4))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H4))) ) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_4874_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xb: A] :
          ( has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D5),topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rr(fun(A,A),fun(A,fun(A,A)),F3),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_4875_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),F4: filter(B),A2: A] :
          ( filterlim(A,B,F3,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_rt(fun(A,B),fun(A,fun(A,B)),F3),A2),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_4876_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B5)) ) ) ) ) ).

% Max.subset_imp
tff(fact_4877_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M4: set(A),N3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M4),N3)
         => ( ( M4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M4)),aa(set(A),A,lattic643756798349783984er_Max(A),N3)) ) ) ) ) ).

% Max_mono
tff(fact_4878_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X3)),aa(A,A,H,Y))
         => ( aa(set(A),$o,finite_finite2(A),N3)
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798349783984er_Max(A),N3)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,H),N3)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_4879_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B5)),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(A),A,lattic643756798349783984er_Max(A),A5) ) ) ) ) ) ).

% Max.subset
tff(fact_4880_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4881_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A5),A5) ) ) ) ) ).

% Max.closed
tff(fact_4882_card__le__Suc__Max,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => aa(nat,$o,aa(nat,fun(nat,$o),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_4883_Sup__nat__def,axiom,
    ! [X5: set(nat)] :
      aa(set(nat),nat,complete_Sup_Sup(nat),X5) = $ite(X5 = bot_bot(set(nat)),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),X5)) ).

% Sup_nat_def
tff(fact_4884_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      divide_divide(nat,Mb,Nb) = $ite(Nb = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ru(nat,fun(nat,fun(nat,$o)),Mb),Nb)))) ).

% divide_nat_def
tff(fact_4885_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S2: set(A),F3: fun(A,B),K2: B] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_rv(fun(A,B),fun(B,fun(A,B)),F3),K2)),S2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F3),S2))),K2) ) ) ) ) ).

% Max_add_commute
tff(fact_4886_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_rw(nat,fun(nat,fun(nat,$o)),Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_4887_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F3: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X3: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S)
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),A2),X3),aa(A,A,F3,X3)) )
           => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_4888_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F3: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X3: A] :
                ( ( X3 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S)
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),A2),X3),aa(A,A,F3,X3)) ) )
           => filterlim(A,A,F3,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_4889_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G3: fun(A,B),G5: filter(B),Xb: A,S2: set(A),F4: filter(B),D3: real,F3: fun(A,B)] :
          ( filterlim(A,B,G3,G5,topolo174197925503356063within(A,Xb,S2))
         => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G5),F4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
             => ( ! [X8: A] :
                    ( member(A,X8,S2)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X8,Xb))
                     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xb)),D3)
                       => ( aa(A,B,F3,X8) = aa(A,B,G3,X8) ) ) ) )
               => filterlim(A,B,F3,F4,topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_4890_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K2: real,F3: fun(nat,real),G3: fun(A,fun(nat,B))] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
         => ( summable(real,F3)
           => ( ! [H4: A,N2: nat] :
                  ( ( H4 != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H4)),K2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G3,H4),N2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F3,N2)),real_V7770717601297561774m_norm(A,H4))) ) )
             => filterlim(A,B,aTP_Lamp_rx(fun(A,fun(nat,B)),fun(A,B),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_4891_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = $ite(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)),insert(A),Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(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)),insert(A),Xb),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_4892_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = $ite(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)),insert(A),Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(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)),insert(A),Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_4893_sum__le__card__Max,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image2(A,nat,F3),A5)))) ) ).

% sum_le_card_Max
tff(fact_4894_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)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(nat,real,A2,zero_zero(nat)))
         => ! [N4: nat] : member(real,suminf(real,aTP_Lamp_ry(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_ry(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_4895_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)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,A2,zero_zero(nat))),zero_zero(real))
         => ! [N4: nat] : member(real,suminf(real,aTP_Lamp_ry(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_ry(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_4896_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),Nb: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2))
       => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_4897_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_4898_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rz(A,fun(fun(nat,A),fun(nat,A)),C3),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_4899_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sa(A,fun(fun(nat,A),fun(nat,A)),C3),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_4900_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sb(A,fun(fun(nat,A),fun(nat,A)),C3),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_4901_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_4902_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ? [U3: fun(nat,A)] :
              ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,U3,N4)),Xb)
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_4903_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ? [U3: fun(nat,A)] :
              ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(nat,A,U3,N4))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_4904_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),K2: nat,A2: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(nat,fun(nat,A)),F3),K2),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_4905_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),A2: A,K2: nat] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(nat,fun(nat,A)),F3),K2),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_4906_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,A2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N7: nat] :
              ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_4907_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,A2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N7: nat] :
              ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,X5,N2)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb) ) ) ) ).

% LIMSEQ_le_const
tff(fact_4908_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,N3: nat,C5: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),aa(nat,A,F3,N2)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),L) ) ) ) ).

% Lim_bounded2
tff(fact_4909_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,M4: nat,C5: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N2)),C5) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C5) ) ) ) ).

% Lim_bounded
tff(fact_4910_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,Y5: fun(nat,A),Yb: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( filterlim(nat,A,Y5,topolo7230453075368039082e_nhds(A,Yb),at_top(nat))
           => ( ? [N7: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,Y5,N2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ) ) ).

% LIMSEQ_le
tff(fact_4911_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N3: nat,X5: fun(nat,A),Y5: fun(nat,A),Xb: A,Yb: A] :
          ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,Y5,N2)) )
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( filterlim(nat,A,Y5,topolo7230453075368039082e_nhds(A,Yb),at_top(nat))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ) ) ).

% lim_mono
tff(fact_4912_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N2: nat] : member(A,aa(nat,A,B2,N2),S)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,complete_Sup_Sup(A),S)) ) ) ) ).

% Sup_lim
tff(fact_4913_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N2: nat] : member(A,aa(nat,A,B2,N2),S)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S)),A2) ) ) ) ).

% Inf_lim
tff(fact_4914_Inf__as__limit,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ? [U3: fun(nat,A)] :
              ( ! [N4: nat] : member(A,aa(nat,A,U3,N4),A5)
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Inf_Inf(A),A5)),at_top(nat)) ) ) ) ).

% Inf_as_limit
tff(fact_4915_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_4916_mult__nat__left__at__top,axiom,
    ! [C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_4917_mult__nat__right__at__top,axiom,
    ! [C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => filterlim(nat,nat,aTP_Lamp_sd(nat,fun(nat,nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_4918_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: fun(nat,A),Xb: A] :
          ( topological_monoseq(A,A2)
         => ( filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N4)),Xb)
                & ! [M3: nat,N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,M3)),aa(nat,A,A2,N4)) ) )
              | ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,A2,N4))
                & ! [M3: nat,N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N4)),aa(nat,A,A2,M3)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_4919_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_se(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_4920_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_sf(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_4921_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X5: fun(nat,A),Xb: A,L: nat] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sg(fun(nat,A),fun(nat,fun(nat,A)),X5),L),topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_4922_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R3),aa(nat,real,X5,N2)) )
     => filterlim(nat,real,aTP_Lamp_sh(fun(nat,real),fun(nat,real),X5),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_4923_LIMSEQ__root__const,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
     => filterlim(nat,real,aTP_Lamp_si(real,fun(nat,real),C3),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_4924_increasing__LIMSEQ,axiom,
    ! [F3: fun(nat,real),L: real] :
      ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N2)),aa(nat,real,F3,aa(nat,nat,suc,N2)))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N2)),L)
       => ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F3,N4)),E2)) )
         => filterlim(nat,real,F3,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_4925_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N),L5)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_4926_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N2),L5)),R3) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_4927_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No3: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N4)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N4),L5)),R2) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_4928_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_sj(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_4929_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => filterlim(nat,real,power_power(real,Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_4930_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_sk(fun(nat,A),fun(nat,A),F3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_4931_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_sl(fun(nat,A),fun(nat,A),F3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),C3)) ) ) ).

% telescope_sums'
tff(fact_4932_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_sm(real,fun(real,fun(nat,real)),Xb),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_4933_LIMSEQ__abs__realpow__zero,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C3)),one_one(real))
     => filterlim(nat,real,power_power(real,aa(real,real,abs_abs(real),C3)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_4934_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C3)),one_one(real))
     => filterlim(nat,real,power_power(real,C3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_4935_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => filterlim(nat,real,aTP_Lamp_sn(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_4936_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A),S: A] :
          ( sums(A,F3,S)
        <=> filterlim(nat,A,aTP_Lamp_so(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_4937_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),Xb: real] :
          ( filterlim(nat,real,aTP_Lamp_sp(fun(nat,A),fun(nat,real),F3),topolo7230453075368039082e_nhds(real,Xb),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => summable(A,F3) ) ) ) ).

% root_test_convergence
tff(fact_4938_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,N)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_4939_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,N2)),L5))),R3) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_4940_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No3: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N4)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,N4)),L5))),R2) ) ) ) ) ).

% LIMSEQ_D
tff(fact_4941_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => filterlim(nat,A,power_power(A,Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_4942_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),No)
                  & ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_4943_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F3: fun(A,nat),F4: filter(A),Xb: B] :
          ( filterlim(A,nat,F3,at_top(nat),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,Xb)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_sq(fun(A,nat),fun(B,fun(A,B)),F3),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_4944_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2))))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_4945_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Df: A,Z: A,S: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F3,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_sr(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F3),Z),S),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_4946_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_ss(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_4947_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xb))
         => filterlim(nat,A,aTP_Lamp_st(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_4948_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_ie(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_4949_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2))
       => ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L3: real] :
              ( ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)))),L3)
              & filterlim(nat,real,aTP_Lamp_su(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat))
              & ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L3),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_sv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_4950_summable__Leibniz_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => filterlim(nat,real,aTP_Lamp_sv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_4951_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2))
       => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2))
         => filterlim(nat,real,aTP_Lamp_sv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_4952_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,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_sw(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F9),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_4953_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),D5: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F3,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D5)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_sx(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F3),D5),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_4954_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,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_sw(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F9),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_within
tff(fact_4955_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ( real_V3181309239436604168linear(A,B,G3)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ph(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% bounded_linear_add
tff(fact_4956_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_pg(A,B)) ) ).

% bounded_linear_zero
tff(fact_4957_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G3: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ( filterlim(C,A,G3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sy(fun(A,B),fun(fun(C,A),fun(C,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_4958_has__derivative__within__singleton__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G3: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F3,G3,topolo174197925503356063within(A,Xb,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))
        <=> real_V3181309239436604168linear(A,B,G3) ) ) ).

% has_derivative_within_singleton_iff
tff(fact_4959_filterlim__pow__at__top,axiom,
    ! [A: $tType,Nb: nat,F3: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,F3,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qi(nat,fun(fun(A,real),fun(A,real)),Nb),F3),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_4960_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_4961_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),K5: real] :
          ( ! [X3: A,Y: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X3)),aa(A,B,F3,Y))
         => ( ! [R3: real,X3: A] : aa(A,B,F3,aa(A,A,real_V8093663219630862766scaleR(A,R3),X3)) = aa(B,B,real_V8093663219630862766scaleR(B,R3),aa(A,B,F3,X3))
           => ( ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K5))
             => real_V3181309239436604168linear(A,B,F3) ) ) ) ) ).

% bounded_linear_intro
tff(fact_4962_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_4963_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ta(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_4964_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F3: fun(A,real),F4: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),S),zero_zero(real))
     => ( filterlim(A,real,F3,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tb(real,fun(fun(A,real),fun(A,real)),S),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_neg_powr
tff(fact_4965_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F3: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),X3)
         => ? [Y3: real] :
              ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),zero_zero(real)) ) )
     => ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F3,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_4966_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,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_tc(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F9),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_4967_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F9: fun(A,B),Xb: A,F3: 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_td(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F9),Xb),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivativeI
tff(fact_4968_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & ? [E4: fun(A,B)] :
                ( ! [H5: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,Xb)),aa(A,B,F9,H5))),aa(A,B,E4,H5))
                & filterlim(A,real,aTP_Lamp_te(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_4969_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E3: real,F9: fun(A,B),S: set(A),Xb: A,F3: fun(A,B),H6: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => ( real_V3181309239436604168linear(A,B,F9)
           => ( ! [Y: A] :
                  ( member(A,Y,S)
                 => ( ( Y != Xb )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,Xb)),E3)
                     => aa(real,$o,aa(real,fun(real,$o),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,F3,Y)),aa(A,B,F3,Xb))),aa(A,B,F9,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Xb)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Xb)))),aa(A,real,H6,Y)) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S))
               => has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_4970_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F9: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F3,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_tg(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F3),F9),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_4971_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xb: A,S2: set(A),F3: fun(A,B),F9: fun(A,B)] :
          ( member(A,Xb,S2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( has_derivative(A,B,F3,F9,topolo174197925503356063within(A,Xb,S2))
            <=> ( real_V3181309239436604168linear(A,B,F9)
                & ? [E4: fun(A,B)] :
                    ( ! [H5: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5),S2)
                       => ( aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,Xb)),aa(A,B,F9,H5))),aa(A,B,E4,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_te(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_4972_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_th(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_4973_open__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1002775350975398744n_open(A,bot_bot(set(A))) ) ).

% open_empty
tff(fact_4974_open__Inter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(set(A))] :
          ( aa(set(set(A)),$o,finite_finite2(set(A)),S2)
         => ( ! [X3: set(A)] :
                ( member(set(A),X3,S2)
               => topolo1002775350975398744n_open(A,X3) )
           => topolo1002775350975398744n_open(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S2)) ) ) ) ).

% open_Inter
tff(fact_4975_open__INT,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A5: set(A),B5: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => topolo1002775350975398744n_open(B,aa(A,set(B),B5,X3)) )
           => topolo1002775350975398744n_open(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A5))) ) ) ) ).

% open_INT
tff(fact_4976_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A5: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xb) )
           => ~ member(A,aa(set(A),A,complete_Sup_Sup(A),A5),A5) ) ) ) ).

% Sup_notin_open
tff(fact_4977_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A5: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X3) )
           => ~ member(A,aa(set(A),A,complete_Inf_Inf(A),A5),A5) ) ) ) ).

% Inf_notin_open
tff(fact_4978_not__open__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xb: A] : ~ topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) ) ).

% not_open_singleton
tff(fact_4979_separation__t2,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,Yb: A] :
          ( ( Xb != Yb )
        <=> ? [U4: set(A),V5: set(A)] :
              ( topolo1002775350975398744n_open(A,U4)
              & topolo1002775350975398744n_open(A,V5)
              & member(A,Xb,U4)
              & member(A,Yb,V5)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),V5) = bot_bot(set(A)) ) ) ) ) ).

% separation_t2
tff(fact_4980_hausdorff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,Yb: A] :
          ( ( Xb != Yb )
         => ? [U5: set(A),V6: set(A)] :
              ( topolo1002775350975398744n_open(A,U5)
              & topolo1002775350975398744n_open(A,V6)
              & member(A,Xb,U5)
              & member(A,Yb,V6)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V6) = bot_bot(set(A)) ) ) ) ) ).

% hausdorff
tff(fact_4981_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,D3: real] : topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_ti(A,fun(real,fun(A,$o)),Xb),D3))) ) ).

% open_ball
tff(fact_4982_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),Xb: A,Yb: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( member(A,Xb,S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B4)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Xb,B4)),S2) ) ) ) ) ) ).

% open_right
tff(fact_4983_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X4: A] :
              ( member(A,X4,S2)
             => ? [E4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
                  & ! [Y4: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y4,X4)),E4)
                     => member(A,Y4,S2) ) ) ) ) ) ).

% open_dist
tff(fact_4984_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,S: set(A)] :
          ( ( topolo174197925503356063within(A,Xb,S) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_tj(A,A)) = Xb ) ) ) ).

% Lim_ident_at
tff(fact_4985_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S9: set(A)] :
              ( topolo1002775350975398744n_open(A,S9)
             => ( member(A,F0,S9)
               => ? [N6: nat] :
                  ! [N: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                   => member(A,aa(nat,A,F3,N),S9) ) ) ) ) ) ).

% lim_explicit
tff(fact_4986_not__tendsto__and__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F4: filter(A),F3: fun(A,B),C3: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
           => ~ filterlim(A,B,F3,at_infinity(B),F4) ) ) ) ).

% not_tendsto_and_filterlim_at_infinity
tff(fact_4987_tendsto__Lim,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Net: filter(A),F3: fun(A,B),L: B] :
          ( ( Net != bot_bot(filter(A)) )
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),Net)
           => ( topolo3827282254853284352ce_Lim(A,B,Net,F3) = L ) ) ) ) ).

% tendsto_Lim
tff(fact_4988_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B),C3: B] :
          ( filterlim(A,B,F3,at_infinity(B),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,C3),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tk(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_4989_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( filterlim(A,B,G3,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tk(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_4990_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A,S2: set(A),T2: set(A),U2: set(A)] :
          ( member(A,Xb,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)),T2),S2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),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)),insert(A),Xb),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,Xb,T2) = topolo174197925503356063within(A,Xb,U2) ) ) ) ) ) ).

% at_within_nhd
tff(fact_4991_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)),insert(A),A2),bot_bot(set(A)))) ) ) ).

% at_eq_bot_iff
tff(fact_4992_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_4993_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( ( C3 != zero_zero(B) )
           => ( filterlim(A,B,G3,at_infinity(B),F4)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tl(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),at_infinity(B),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_4994_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( filterlim(A,B,G3,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_4995_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F3,at_infinity(B),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_tn(fun(A,B),fun(nat,fun(A,B)),F3),Nb),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_4996_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_4997_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_qo(fun(A,B),fun(A,B),G3),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G3,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_4998_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S2: set(B),F3: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( member(B,A2,S2)
           => ( topolo1002775350975398744n_open(B,S2)
             => ( filterlim(B,C,F3,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,S2))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_rn(B,fun(fun(B,C),fun(B,C)),A2),F3),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_4999_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),C3: A,F4: filter(A),G3: fun(A,A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,C3),F4)
         => ( filterlim(A,A,G3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C3 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_nz(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_5000_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xb))
         => filterlim(nat,A,power_power(A,Xb),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_5001_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F3: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F3,at_bot(real),F4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_to(nat,fun(fun(real,real),fun(real,real)),Nb),F3),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_5002_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C3: fun(nat,A),K2: nat,Nb: nat,B5: real] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_tp(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C3),Nb),B5),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_5003_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F3: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F3,at_bot(real),F4)
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_to(nat,fun(fun(real,real),fun(real,real)),Nb),F3),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_5004_eventually__sequentially__seg,axiom,
    ! [P2: fun(nat,$o),K2: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_tq(fun(nat,$o),fun(nat,fun(nat,$o)),P2),K2),at_top(nat))
    <=> eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_5005_eventually__const,axiom,
    ! [A: $tType,F4: filter(A),P2: $o] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_ni($o,fun(A,$o),(P2)),F4)
      <=> (P2) ) ) ).

% eventually_const
tff(fact_5006_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F3,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_tr(fun(A,real),fun(A,$o),F3),F4)
       => filterlim(A,real,F3,at_bot(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_5007_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_ts(A,fun(A,$o)),C3),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_5008_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_bot(A))
        <=> ? [N6: A] :
            ! [N: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N),N6)
             => aa(A,$o,P2,N) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_5009_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C3: A] : eventually(A,aTP_Lamp_tt(A,fun(A,$o),C3),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_5010_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_bot(A))
        <=> ? [N6: A] :
            ! [N: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N),N6)
             => aa(A,$o,P2,N) ) ) ) ).

% eventually_at_bot_dense
tff(fact_5011_eventually__happens_H,axiom,
    ! [A: $tType,F4: filter(A),P2: fun(A,$o)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,P2,F4)
       => ? [X_12: A] : aa(A,$o,P2,X_12) ) ) ).

% eventually_happens'
tff(fact_5012_eventually__happens,axiom,
    ! [A: $tType,P2: fun(A,$o),Net: filter(A)] :
      ( eventually(A,P2,Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_12: A] : aa(A,$o,P2,X_12) ) ) ).

% eventually_happens
tff(fact_5013_eventually__bot,axiom,
    ! [A: $tType,P2: fun(A,$o)] : eventually(A,P2,bot_bot(filter(A))) ).

% eventually_bot
tff(fact_5014_trivial__limit__def,axiom,
    ! [A: $tType,F4: filter(A)] :
      ( ( F4 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_al(A,$o),F4) ) ).

% trivial_limit_def
tff(fact_5015_eventually__const__iff,axiom,
    ! [A: $tType,P2: $o,F4: filter(A)] :
      ( eventually(A,aTP_Lamp_ni($o,fun(A,$o),(P2)),F4)
    <=> ( (P2)
        | ( F4 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_5016_False__imp__not__eventually,axiom,
    ! [A: $tType,P2: fun(A,$o),Net: filter(A)] :
      ( ! [X3: A] : ~ aa(A,$o,P2,X3)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P2,Net) ) ) ).

% False_imp_not_eventually
tff(fact_5017_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_tu(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_5018_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z6),C3)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_tv(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_5019_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_tv(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ).

% filterlim_at_bot
tff(fact_5020_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z6),C3)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_tw(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_5021_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,P2: fun(A,$o)] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X3)
             => aa(A,$o,P2,X3) )
         => eventually(A,P2,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_5022_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_top(A))
        <=> ? [N6: A] :
            ! [N: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N6),N)
             => aa(A,$o,P2,N) ) ) ) ).

% eventually_at_top_linorder
tff(fact_5023_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_top(A))
        <=> ? [N6: A] :
            ! [N: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N6),N)
             => aa(A,$o,P2,N) ) ) ) ).

% eventually_at_top_dense
tff(fact_5024_eventually__sequentiallyI,axiom,
    ! [C3: nat,P2: fun(nat,$o)] :
      ( ! [X3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C3),X3)
         => aa(nat,$o,P2,X3) )
     => eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_5025_eventually__sequentially,axiom,
    ! [P2: fun(nat,$o)] :
      ( eventually(nat,P2,at_top(nat))
    <=> ? [N6: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
         => aa(nat,$o,P2,N) ) ) ).

% eventually_sequentially
tff(fact_5026_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_5027_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),ord_less_eq(A),C3),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_5028_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),C3),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_5029_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F4),at_top(nat))
    <=> ! [N6: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),F4) ) ).

% le_sequentially
tff(fact_5030_sequentially__offset,axiom,
    ! [P2: fun(nat,$o),K2: nat] :
      ( eventually(nat,P2,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_tq(fun(nat,$o),fun(nat,fun(nat,$o)),P2),K2),at_top(nat)) ) ).

% sequentially_offset
tff(fact_5031_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_tr(fun(A,real),fun(A,$o),F3),F4)
       => filterlim(A,real,aTP_Lamp_tx(fun(A,real),fun(A,real),F3),at_bot(real),F4) ) ) ).

% filterlim_inverse_at_bot
tff(fact_5032_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),top_top(A))
         => ( eventually(A,P2,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),top_top(A))
                & ! [Z5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Z5)
                   => aa(A,$o,P2,Z5) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_5033_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F3: fun(A,B),P2: fun(B,$o),G3: fun(B,A)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P2,X3)
               => ( aa(A,B,F3,aa(B,A,G3,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P2,X3)
                 => aa(A,$o,Q,aa(B,A,G3,X3)) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P2,at_top(B))
                 => filterlim(A,B,F3,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_5034_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P2: fun(A,$o),Xb: A] :
          ( eventually(A,P2,topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_lessThan(A),Xb)))
        <=> ? [B7: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Xb)
              & ! [Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Y4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xb)
                   => aa(A,$o,P2,Y4) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_5035_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Yb: A,Xb: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( eventually(A,P2,topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_lessThan(A),Xb)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Xb)
                & ! [Y4: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Y4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xb)
                     => aa(A,$o,P2,Y4) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_5036_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Net: filter(A),H: fun(A,B),C3: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ty(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ty(fun(A,B),fun(fun(A,B),fun(A,$o)),G3),H),Net)
           => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),Net)
             => ( filterlim(A,B,H,topolo7230453075368039082e_nhds(B,C3),Net)
               => filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,C3),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_5037_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Xb: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xb),F4)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),Xb)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(B,fun(A,$o)),F3),L4),F4) )
            & ! [U6: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xb),U6)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,$o)),F3),U6),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_5038_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Yb: A,F3: fun(B,A),F4: filter(B)] :
          ( ! [A4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),Yb)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ub(fun(B,A),fun(A,fun(B,$o)),F3),A4),F4) )
         => ( ! [A4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),A4)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_uc(fun(B,A),fun(A,fun(B,$o)),F3),A4),F4) )
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,Yb),F4) ) ) ) ).

% order_tendstoI
tff(fact_5039_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Yb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Yb),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),Yb)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_5040_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Yb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Yb),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Yb),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_5041_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,at_top(B),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ud(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),F4)
           => filterlim(A,B,G3,at_top(B),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_5042_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C3),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ue(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_5043_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ue(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ).

% filterlim_at_top
tff(fact_5044_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_uf(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_5045_eventually__Inf__base,axiom,
    ! [A: $tType,B5: set(filter(A)),P2: fun(A,$o)] :
      ( ( B5 != bot_bot(set(filter(A))) )
     => ( ! [F5: filter(A)] :
            ( member(filter(A),F5,B5)
           => ! [G4: filter(A)] :
                ( member(filter(A),G4,B5)
               => ? [X: filter(A)] :
                    ( member(filter(A),X,B5)
                    & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),X),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),G4)) ) ) )
       => ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5))
        <=> ? [X4: filter(A)] :
              ( member(filter(A),X4,B5)
              & eventually(A,P2,X4) ) ) ) ) ).

% eventually_Inf_base
tff(fact_5046_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P2: fun(B,$o),F4: fun(A,filter(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),A5)))
      <=> ? [Q7: fun(A,fun(B,$o))] :
            ( ! [X4: A] :
                ( member(A,X4,A5)
               => eventually(B,aa(A,fun(B,$o),Q7,X4),aa(A,filter(B),F4,X4)) )
            & ! [Y4: B] :
                ( ! [X4: A] :
                    ( member(A,X4,A5)
                   => aa(B,$o,aa(A,fun(B,$o),Q7,X4),Y4) )
               => aa(B,$o,P2,Y4) ) ) ) ) ).

% eventually_INF_finite
tff(fact_5047_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),A2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_ug(real,fun(real,fun(real,$o)),B2),A2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ).

% eventually_at_left_real
tff(fact_5048_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P2: fun(A,$o),A2: A,S2: set(A)] :
          ( eventually(A,P2,topolo174197925503356063within(A,A2,S2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X4: A] :
                  ( member(A,X4,S2)
                 => ( ( ( X4 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D6) )
                   => aa(A,$o,P2,X4) ) ) ) ) ) ).

% eventually_at
tff(fact_5049_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P2: fun(A,$o),A2: A] :
          ( eventually(A,P2,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X4: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D6)
                 => aa(A,$o,P2,X4) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_5050_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P2: fun(A,$o)] :
          ( ! [X3: A] :
              ( member(A,X3,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P2,X3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P2,topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_5051_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,$o),A2: A] :
          ( eventually(A,P2,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_uh(fun(A,$o),fun(A,fun(A,$o)),P2),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_5052_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F3: fun(A,B),F4: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ui(B,fun(fun(A,B),fun(A,$o)),L),F3),F4)
         => ( ! [X3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X3)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,$o)),F3),X3),F4) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_5053_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_uj(fun(A,B),fun(B,fun(A,$o)),F3),L),F4)
         => ( ! [X3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(B,fun(A,$o)),F3),X3),F4) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_5054_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C3),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_uk(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_5055_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F4: filter(A),F3: fun(A,B),Xb: B,G3: fun(A,B),Yb: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xb),F4)
           => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,Yb),F4)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ul(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),F4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Yb),Xb) ) ) ) ) ) ).

% tendsto_le
tff(fact_5056_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: fun(A,B),Xb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_um(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),Xb) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_5057_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: fun(A,B),Xb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_un(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),A2) ) ) ) ) ).

% tendsto_upperbound
tff(fact_5058_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F3,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_uo(fun(A,real),fun(A,$o),F3),F4)
       => filterlim(A,real,F3,at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_5059_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F4: fun(B,filter(A)),B5: set(B)] :
      ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),B5)))
    <=> ? [X9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),B5)
          & aa(set(B),$o,finite_finite2(B),X9)
          & eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),X9))) ) ) ).

% eventually_INF
tff(fact_5060_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P2: fun(A,$o),A2: A,S2: set(A)] :
          ( eventually(A,P2,topolo174197925503356063within(A,A2,S2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X4: A] :
                  ( member(A,X4,S2)
                 => ( ( ( X4 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X4,A2)),D6) )
                   => aa(A,$o,P2,X4) ) ) ) ) ) ).

% eventually_at_le
tff(fact_5061_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_infinity(A))
        <=> ? [B7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B7)
              & ! [X4: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X4))
                 => aa(A,$o,P,X4) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_5062_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,$o)),F3),L5),F4)
           => filterlim(A,B,F3,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_lessThan(B),L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_5063_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A),E3: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_up(fun(A,B),fun(B,fun(real,fun(A,$o))),F3),L),E3),F4) ) ) ) ).

% tendstoD
tff(fact_5064_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_up(fun(A,B),fun(B,fun(real,fun(A,$o))),F3),L),E2),F4) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% tendstoI
tff(fact_5065_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_up(fun(A,B),fun(B,fun(real,fun(A,$o))),F3),L),E4),F4) ) ) ) ).

% tendsto_iff
tff(fact_5066_eventually__Inf,axiom,
    ! [A: $tType,P2: fun(A,$o),B5: set(filter(A))] :
      ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5))
    <=> ? [X9: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X9),B5)
          & aa(set(filter(A)),$o,finite_finite2(filter(A)),X9)
          & eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X9)) ) ) ).

% eventually_Inf
tff(fact_5067_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F3: fun(A,B),P2: fun(B,$o),G3: fun(B,A),A2: A] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P2,X3)
               => ( aa(A,B,F3,aa(B,A,G3,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P2,X3)
                 => aa(A,$o,Q,aa(B,A,G3,X3)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2)))
               => ( ! [B4: A] :
                      ( aa(A,$o,Q,B4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A2) )
                 => ( eventually(B,P2,at_top(B))
                   => filterlim(A,B,F3,at_top(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_5068_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B5: set(A),F4: fun(A,filter(B)),P2: fun(B,$o)] :
      ( ( B5 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( member(A,A4,B5)
           => ! [B4: A] :
                ( member(A,B4,B5)
               => ? [X: A] :
                    ( member(A,X,B5)
                    & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X)),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,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B5)))
        <=> ? [X4: A] :
              ( member(A,X4,B5)
              & eventually(B,P2,aa(A,filter(B),F4,X4)) ) ) ) ) ).

% eventually_INF_base
tff(fact_5069_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,C),K5: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_uq(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F3),G3),K5),F4)
           => filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_5070_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),A5: set(B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ur(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F3),C3),A5),F4)
           => filterlim(A,B,F3,topolo174197925503356063within(B,C3,A5),F4) ) ) ) ).

% filterlim_at_withinI
tff(fact_5071_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_5072_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C3: real,F3: fun(A,B),F4: filter(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C3)
         => ( filterlim(A,B,F3,at_infinity(B),F4)
          <=> ! [R5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),R5)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_us(fun(A,B),fun(real,fun(A,$o)),F3),R5),F4) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_5073_tendsto__zero__powrI,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_ut(fun(A,real),fun(A,$o),F3),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uu(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_5074_tendsto__powr2,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_ut(fun(A,real),fun(A,$o),F3),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uu(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ) ).

% tendsto_powr2
tff(fact_5075_tendsto__powr_H,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( ( ( A2 != zero_zero(real) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
              & eventually(A,aTP_Lamp_ut(fun(A,real),fun(A,$o),F3),F4) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uu(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ).

% tendsto_powr'
tff(fact_5076_LIM__at__top__divide,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => ( eventually(A,aTP_Lamp_uo(fun(A,real),fun(A,$o),G3),F4)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uv(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_5077_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_uo(fun(A,real),fun(A,$o),F3),F4)
       => filterlim(A,real,aTP_Lamp_tx(fun(A,real),fun(A,real),F3),at_top(real),F4) ) ) ).

% filterlim_inverse_at_top
tff(fact_5078_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_uo(fun(A,real),fun(A,$o),F3),F4)
     => ( filterlim(A,real,aTP_Lamp_tx(fun(A,real),fun(A,real),F3),at_top(real),F4)
      <=> filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_5079_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_5080_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F3: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
         => ? [Y3: real] :
              ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y3) ) )
     => ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F3,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_5081_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_uw(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F3),G3),at_top(nat))
         => ( filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_Cauchy'
tff(fact_5082_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F3,F4)
        <=> ? [Y4: B,K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ux(fun(A,B),fun(B,fun(real,fun(A,$o))),F3),Y4),K6),F4) ) ) ) ).

% Bfun_metric_def
tff(fact_5083_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F3,F4)
        <=> ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_uy(fun(A,B),fun(real,fun(A,$o)),F3),K6),F4) ) ) ) ).

% Bfun_def
tff(fact_5084_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_uz(fun(nat,A),fun(A,fun(nat,A)),F3),C3),at_top(nat))
        <=> bfun(nat,A,F3,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_5085_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( bfun(nat,A,F3,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_uz(fun(nat,A),fun(A,fun(nat,A)),F3),C3),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_5086_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),K2: nat] :
          ( bfun(nat,A,X5,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_va(fun(nat,A),fun(nat,fun(nat,A)),X5),K2),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_5087_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),K2: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_va(fun(nat,A),fun(nat,fun(nat,A)),X5),K2),at_top(nat))
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_5088_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_top(A))
         => eventually(A,aTP_Lamp_vb(fun(A,$o),fun(A,$o),P2),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_5089_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),D3: B] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_vc(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A5),B5),D3))) ) ) ).

% finite_set_of_finite_funs
tff(fact_5090_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_et(A,fun(fun(nat,A),fun(nat,A)),C3),F3),at_top(nat))
          <=> bfun(nat,A,F3,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_5091_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N4))),K8) ) ) ) ).

% BseqD
tff(fact_5092_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => ~ ! [K8: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
               => ~ ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N4))),K8) ) ) ) ).

% BseqE
tff(fact_5093_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K5: real,X5: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
         => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N2))),K5)
           => bfun(nat,A,X5,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_5094_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K6) ) ) ) ).

% Bseq_def
tff(fact_5095_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [N6: nat] :
            ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% Bseq_iff1a
tff(fact_5096_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [X4: A] :
                ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X5,N)),aa(A,A,uminus_uminus(A),X4)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_5097_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [N6: nat] :
                ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X5,N)),aa(A,A,uminus_uminus(A),aa(nat,A,X5,N6))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_5098_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_qo(fun(A,B),fun(A,B),F3),F4) ) ) ) ).

% Bfun_inverse
tff(fact_5099_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F3,F4)
         => ~ ! [B8: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B8)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_uy(fun(A,B),fun(real,fun(A,$o)),F3),B8),F4) ) ) ) ).

% BfunE
tff(fact_5100_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_vd(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F3),G3),at_top(nat))
         => ( filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_bounded_partials
tff(fact_5101_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o)] : order_Greatest(A,P2) = the(A,aTP_Lamp_ve(fun(A,$o),fun(A,$o),P2)) ) ).

% Greatest_def
tff(fact_5102_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F4: filter(A)] :
          ( topolo6773858410816713723filter(A,F4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [P5: fun(A,$o)] :
                  ( eventually(A,P5,F4)
                  & ! [X4: A,Y4: A] :
                      ( ( aa(A,$o,P5,X4)
                        & aa(A,$o,P5,Y4) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,Y4)),E4) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_5103_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or3652927894154168847AtMost(nat,L,U)) ).

% finite_greaterThanAtMost
tff(fact_5104_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( member(A,I2,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_5105_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K2)
         => ( set_or3652927894154168847AtMost(A,K2,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_5106_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K2,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_5107_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K2,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_5108_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioc_iff
tff(fact_5109_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C3: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C3)),set_or3652927894154168847AtMost(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ).

% image_add_greaterThanAtMost
tff(fact_5110_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Yb,Xb)) = Xb ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_5111_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Xb,Yb)) = Yb ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_5112_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Yb,Xb)) = Yb ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_5113_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Xb,Yb)) = Xb ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_5114_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or3652927894154168847AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_greaterThanAtMost
tff(fact_5115_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C3)),set_or3652927894154168847AtMost(A,A2,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_5116_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C3)),set_or7035219750837199246ssThan(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)) ) ).

% image_diff_atLeastLessThan
tff(fact_5117_image__uminus__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Yb: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or7035219750837199246ssThan(A,Xb,Yb)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Yb),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_atLeastLessThan
tff(fact_5118_image__uminus__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Yb: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or3652927894154168847AtMost(A,Xb,Yb)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Yb),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_greaterThanAtMost
tff(fact_5119_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C3,D3) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C3) )
            | ( ( A2 = C3 )
              & ( B2 = D3 ) ) ) ) ) ).

% Ioc_inj
tff(fact_5120_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_5121_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_5122_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A2,B2)) ) ) ).

% infinite_Ioc
tff(fact_5123_ivl__disj__int__two_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,Mb: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or3652927894154168847AtMost(A,Mb,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(6)
tff(fact_5124_GreatestI__ex__nat,axiom,
    ! [P2: fun(nat,$o),B2: nat] :
      ( ? [X_1: nat] : aa(nat,$o,P2,X_1)
     => ( ! [Y: nat] :
            ( aa(nat,$o,P2,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),B2) )
       => aa(nat,$o,P2,order_Greatest(nat,P2)) ) ) ).

% GreatestI_ex_nat
tff(fact_5125_Greatest__le__nat,axiom,
    ! [P2: fun(nat,$o),K2: nat,B2: nat] :
      ( aa(nat,$o,P2,K2)
     => ( ! [Y: nat] :
            ( aa(nat,$o,P2,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),order_Greatest(nat,P2)) ) ) ).

% Greatest_le_nat
tff(fact_5126_GreatestI__nat,axiom,
    ! [P2: fun(nat,$o),K2: nat,B2: nat] :
      ( aa(nat,$o,P2,K2)
     => ( ! [Y: nat] :
            ( aa(nat,$o,P2,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),B2) )
       => aa(nat,$o,P2,order_Greatest(nat,P2)) ) ) ).

% GreatestI_nat
tff(fact_5127_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D3)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),A2) ) ) ) ).

% Ioc_disjoint
tff(fact_5128_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),Xb: A,Yb: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( member(A,Xb,S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),Xb)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B4,Xb)),S2) ) ) ) ) ) ).

% open_left
tff(fact_5129_ivl__disj__int__two_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,Mb: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or3652927894154168847AtMost(A,Mb,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(8)
tff(fact_5130_ivl__disj__int__one_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(3)
tff(fact_5131_ivl__disj__int__two_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,Mb: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or5935395276787703475ssThan(A,Mb,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(2)
tff(fact_5132_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o),Xb: A] :
          ( aa(A,$o,P2,Xb)
         => ( ! [Y: A] :
                ( aa(A,$o,P2,Y)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) )
           => ( order_Greatest(A,P2) = Xb ) ) ) ) ).

% Greatest_equality
tff(fact_5133_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P2,Xb)
         => ( ! [Y: A] :
                ( aa(A,$o,P2,Y)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) )
           => ( ! [X3: A] :
                  ( aa(A,$o,P2,X3)
                 => ( ! [Y3: A] :
                        ( aa(A,$o,P2,Y3)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X3) )
                   => aa(A,$o,Q,X3) ) )
             => aa(A,$o,Q,order_Greatest(A,P2)) ) ) ) ) ).

% GreatestI2_order
tff(fact_5134_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.head
tff(fact_5135_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.head
tff(fact_5136_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_5137_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_5138_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_5139_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)),insert(A),A2),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_5140_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5141_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),R: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P2,R) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_vg(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),P2),R)))) ).

% same_fst_def
tff(fact_5142_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),P2: fun(A,$o)] : aa(fun(A,$o),A,least(A,Less_eq),P2) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_vh(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),P2)) ).

% ord.Least_def
tff(fact_5143_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P2: fun(A,$o),F3: fun(A,B),A5: B,B5: set(B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,topolo174197925503356063within(B,A5,B5)))
        <=> ? [S9: set(B)] :
              ( topolo1002775350975398744n_open(B,S9)
              & member(B,A5,S9)
              & ! [X4: A] :
                  ( member(B,aa(A,B,F3,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S9),B5)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A5),bot_bot(set(B)))))
                 => aa(A,$o,P2,X4) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_5144_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),$o,finite_finite2(int),set_or3652927894154168847AtMost(int,L,U)) ).

% finite_greaterThanAtMost_int
tff(fact_5145_filtercomap__bot,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : filtercomap(A,B,F3,bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtercomap_bot
tff(fact_5146_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or3652927894154168847AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_greaterThanAtMost_int
tff(fact_5147_same__fstI,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),Xb: A,Y6: B,Yb: B,R: fun(A,set(product_prod(B,B)))] :
      ( aa(A,$o,P2,Xb)
     => ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y6),Yb),aa(A,set(product_prod(B,B)),R,Xb))
       => member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y6)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb)),same_fst(A,B,P2,R)) ) ) ).

% same_fstI
tff(fact_5148_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_5149_filtercomap__neq__bot,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F3: fun(B,A)] :
      ( ! [P6: fun(A,$o)] :
          ( eventually(A,P6,F4)
         => ? [X: B] : aa(A,$o,P6,aa(B,A,F3,X)) )
     => ( filtercomap(B,A,F3,F4) != bot_bot(filter(B)) ) ) ).

% filtercomap_neq_bot
tff(fact_5150_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_5151_eventually__filtercomap__at__top__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P2: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,at_top(B)))
        <=> ? [N6: B] :
            ! [X4: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N6),aa(A,B,F3,X4))
             => aa(A,$o,P2,X4) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_5152_eventually__filtercomap__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [P2: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,at_top(B)))
        <=> ? [N6: B] :
            ! [X4: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N6),aa(A,B,F3,X4))
             => aa(A,$o,P2,X4) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_5153_filtercomap__neq__bot__surj,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F3: fun(B,A)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
       => ( filtercomap(B,A,F3,F4) != bot_bot(filter(B)) ) ) ) ).

% filtercomap_neq_bot_surj
tff(fact_5154_eventually__filtercomap__at__bot__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P2: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,at_bot(B)))
        <=> ? [N6: B] :
            ! [X4: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),N6)
             => aa(A,$o,P2,X4) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_5155_eventually__filtercomap__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [P2: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,at_bot(B)))
        <=> ? [N6: B] :
            ! [X4: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),N6)
             => aa(A,$o,P2,X4) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_5156_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_ts(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_5157_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A,S: set(A)] : topolo174197925503356063within(A,Xb,S) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image2(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_vi(A,fun(set(A),fun(set(A),filter(A))),Xb),S)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_vj(A,fun(set(A),$o),Xb)))) ) ).

% at_within_eq
tff(fact_5158_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_oh(fun(nat,A),fun(A,A),C3)) ) ) ) ).

% isCont_powser
tff(fact_5159_continuous__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B)] : topolo3448309680560233919inuous(A,B,bot_bot(filter(A)),F3) ) ).

% continuous_bot
tff(fact_5160_continuous__trivial__limit,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Net: filter(A),F3: fun(A,B)] :
          ( ( Net = bot_bot(filter(A)) )
         => topolo3448309680560233919inuous(A,B,Net,F3) ) ) ).

% continuous_trivial_limit
tff(fact_5161_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,C,F4,G3)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F4,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vk(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% continuous_Pair
tff(fact_5162_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F3: fun(A,B),G3: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => 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_vk(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% isCont_Pair
tff(fact_5163_continuous__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vl(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_max
tff(fact_5164_linorder_OMin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : lattices_Min(A,Less_eq) = lattices_Min(A,Less_eq) ).

% linorder.Min.cong
tff(fact_5165_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vm(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_add
tff(fact_5166_principal__eq__bot__iff,axiom,
    ! [A: $tType,X5: set(A)] :
      ( ( principal(A,X5) = bot_bot(filter(A)) )
    <=> ( X5 = bot_bot(set(A)) ) ) ).

% principal_eq_bot_iff
tff(fact_5167_bot__eq__principal__empty,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = principal(A,bot_bot(set(A))) ).

% bot_eq_principal_empty
tff(fact_5168_IVT,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F3: fun(B,A),A2: B,Yb: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,A2)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),aa(B,A,F3,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( ! [X3: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X3)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X3,top_top(set(B))),F3) )
               => ? [X3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X3)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B2)
                    & ( aa(B,A,F3,X3) = Yb ) ) ) ) ) ) ) ).

% IVT
tff(fact_5169_IVT2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F3: fun(B,A),B2: B,Yb: A,A2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,B2)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),aa(B,A,F3,A2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( ! [X3: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X3)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X3,top_top(set(B))),F3) )
               => ? [X3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X3)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B2)
                    & ( aa(B,A,F3,X3) = Yb ) ) ) ) ) ) ) ).

% IVT2
tff(fact_5170_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),G3)
           => ( ( aa(A,B,G3,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_5171_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A2: A,F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_vm(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% isCont_add
tff(fact_5172_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A2: A,S: set(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F3)
         => ( ( aa(A,B,F3,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_vo(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_5173_nhds__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [Xb: A] : topolo7230453075368039082e_nhds(A,Xb) = principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) ) ).

% nhds_discrete
tff(fact_5174_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,S: set(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F3)
         => ( ( aa(A,B,F3,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_vp(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_5175_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => ( ( aa(A,B,G3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_divide
tff(fact_5176_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F4: filter(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( ( aa(A,B,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_vo(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_inverse
tff(fact_5177_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( ( aa(A,B,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_vp(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_sgn
tff(fact_5178_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M7),aa(real,A,F3,X)) )
                & ? [X3: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
                    & ( aa(real,A,F3,X3) = M7 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_5179_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F3,X)),M7) )
                & ? [X3: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
                    & ( aa(real,A,F3,X3) = M7 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_5180_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M7: A] :
              ! [X: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F3,X)),M7) ) ) ) ) ).

% isCont_bounded
tff(fact_5181_isCont__inverse__function2,axiom,
    ! [A2: real,Xb: real,B2: real,G3: fun(real,real),F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ( ! [Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z2),B2)
               => ( aa(real,real,G3,aa(real,real,F3,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z2),B2)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F3) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F3,Xb),top_top(set(real))),G3) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_5182_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => ( ( aa(A,B,G3,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% isCont_divide
tff(fact_5183_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( ( aa(A,B,F3,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_vp(fun(A,B),fun(A,B),F3)) ) ) ) ).

% isCont_sgn
tff(fact_5184_tendsto__principal__singleton,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F3: fun(A,B),Xb: A] : filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,Xb)),principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))) ) ).

% tendsto_principal_singleton
tff(fact_5185_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S: set(A),F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),F3)
         => ( ( cos(A,aa(A,A,F3,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_re(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_within_tan
tff(fact_5186_nhds__discrete__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A] :
          ( topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))
         => ( topolo7230453075368039082e_nhds(A,Xb) = principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) ) ) ) ).

% nhds_discrete_open
tff(fact_5187_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S: set(A),F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),F3)
         => ( ( sin(A,aa(A,A,F3,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_qm(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_within_cot
tff(fact_5188_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Xb: A,A5: set(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A5),F3)
         => ( ( cosh(B,aa(A,B,F3,Xb)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A5),aTP_Lamp_vq(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_5189_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F3)
         => ( ( cos(A,aa(A,A,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_vr(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_re(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_tan
tff(fact_5190_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F3)
         => ( ( sin(A,aa(A,A,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_vr(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_qm(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_cot
tff(fact_5191_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( ( cosh(B,aa(A,B,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_vq(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_tanh
tff(fact_5192_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F3,X)),M7) )
                & ! [N7: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),M7)
                   => ? [X3: real] :
                        ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),aa(real,A,F3,X3)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_5193_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_vs(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_arcosh
tff(fact_5194_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_5195_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_5196_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cosh(A,Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_5197_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( ( cos(A,aa(A,A,F3,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_re(fun(A,A),fun(A,A),F3)) ) ) ) ).

% isCont_tan'
tff(fact_5198_isCont__arcosh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_5199_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S: set(A),F3: fun(A,real),G3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),F3)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),G3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,A2))
             => ( ( aa(A,real,F3,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),aa(fun(A,real),fun(A,real),aTP_Lamp_vt(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_5200_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F3: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( ( sin(A,aa(A,A,F3,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_qm(fun(A,A),fun(A,A),F3)) ) ) ) ).

% isCont_cot'
tff(fact_5201_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F3: fun(A,real),G3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F3)
         => ( topolo3448309680560233919inuous(A,real,F4,G3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A))))
             => ( ( aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_vt(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_5202_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),F3)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vu(A,fun(fun(A,B),fun(A,B)),Xb),F3),topolo7230453075368039082e_nhds(B,aa(A,B,F3,Xb)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_5203_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I6: set(A),F4: fun(A,set(B)),F3: fun(B,C),G5: fun(D,set(C)),J4: set(D)] :
      ( ( I6 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( member(A,I3,I6)
           => ! [J2: A] :
                ( member(A,J2,I6)
               => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,I3)),aa(A,set(B),F4,J2))
                  | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,J2)),aa(A,set(B),F4,I3)) ) ) )
       => ( filterlim(B,C,F3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_vv(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_vw(fun(A,set(B)),fun(A,filter(B)),F4)),I6)))
        <=> ! [X4: D] :
              ( member(D,X4,J4)
             => ? [Xa2: A] :
                  ( member(A,Xa2,I6)
                  & ! [Xb4: B] :
                      ( member(B,Xb4,aa(A,set(B),F4,Xa2))
                     => member(C,aa(B,C,F3,Xb4),aa(D,set(C),G5,X4)) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_5204_DERIV__inverse__function,axiom,
    ! [F3: fun(real,real),D5: real,G3: fun(real,real),Xb: real,A2: real,B2: real] :
      ( has_field_derivative(real,F3,D5,topolo174197925503356063within(real,aa(real,real,G3,Xb),top_top(set(real))))
     => ( ( D5 != zero_zero(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
           => ( ! [Y: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),B2)
                   => ( aa(real,real,F3,aa(real,real,G3,Y)) = Y ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),G3)
               => has_field_derivative(real,G3,aa(real,real,inverse_inverse(real),D5),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_5205_isCont__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_5206_isCont__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_5207_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_vw(fun(A,set(B)),fun(A,filter(B)),F3)),X5)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),X5))) ) ) ).

% INF_principal_finite
tff(fact_5208_LIM__less__bound,axiom,
    ! [B2: real,Xb: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),Xb)
     => ( ! [X3: real] :
            ( member(real,X3,set_or5935395276787703475ssThan(real,B2,Xb))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F3,X3)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),F3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F3,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_5209_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F3: fun(A,real),G3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,A2))
             => ( ( aa(A,real,F3,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_vt(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_5210_isCont__artanh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_5211_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)),insert(A),A2),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_5212_isCont__inverse__function,axiom,
    ! [D3: real,Xb: real,G3: fun(real,real),F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
     => ( ! [Z2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z2),Xb))),D3)
           => ( aa(real,real,G3,aa(real,real,F3,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z2),Xb))),D3)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F3) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F3,Xb),top_top(set(real))),G3) ) ) ) ).

% isCont_inverse_function
tff(fact_5213_GMVT_H,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),G3: fun(real,real),G6: fun(real,real),F9: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [Z2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z2),B2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F3) ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z2),B2)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),G3) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B2)
                 => has_field_derivative(real,G3,aa(real,real,G6,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B2)
                   => has_field_derivative(real,F3,aa(real,real,F9,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),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,F3,B2)),aa(real,real,F3,A2))),aa(real,real,G6,C2)) = 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,G3,B2)),aa(real,real,G3,A2))),aa(real,real,F9,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_5214_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F3: fun(A,B),G3: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( filterlim(B,C,G3,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F3,A2),top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D4) )
                     => ( aa(A,B,F3,X3) != aa(A,B,F3,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rm(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_5215_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
         => ( topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_lessThan(A),Xb)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_vx(A,fun(A,filter(A)),Xb)),aa(A,set(A),set_ord_lessThan(A),Xb))) ) ) ) ).

% at_left_eq
tff(fact_5216_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F3: fun(A,B),G3: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( filterlim(B,C,G3,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F3,A2),top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),D4) )
                     => ( aa(A,B,F3,X3) != aa(A,B,F3,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rq(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_5217_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A2: A,F3: fun(A,B),C3: fun(nat,B),K5: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_vy(fun(nat,B),fun(B,fun(nat,B)),C3),K5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,A2))),real_V7770717601297561774m_norm(B,K5))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_wa(fun(A,B),fun(fun(nat,B),fun(A,B)),F3),C3)) ) ) ) ) ).

% isCont_powser'
tff(fact_5218_complete__uniform,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A)] :
          ( topolo2479028161051973599mplete(A,S2)
        <=> ! [F10: filter(A)] :
              ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F10),principal(A,S2))
             => ( ( F10 != bot_bot(filter(A)) )
               => ( topolo6773858410816713723filter(A,F10)
                 => ? [X4: A] :
                      ( member(A,X4,S2)
                      & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F10),topolo7230453075368039082e_nhds(A,X4)) ) ) ) ) ) ) ).

% complete_uniform
tff(fact_5219_at__within__order,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,S: set(A)] :
          ( ( top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) )
         => ( topolo174197925503356063within(A,Xb,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_wb(A,fun(set(A),fun(A,filter(A))),Xb),S)),aa(A,set(A),set_ord_greaterThan(A),Xb)))),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_wc(A,fun(set(A),fun(A,filter(A))),Xb),S)),aa(A,set(A),set_ord_lessThan(A),Xb)))) ) ) ) ).

% at_within_order
tff(fact_5220_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F3),A5)),B5)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F3)),finite_Fpow(B,A5))),finite_Fpow(A,B5)) ) ).

% image_Fpow_mono
tff(fact_5221_greaterThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,set(A),set_ord_greaterThan(A),Xb) = aa(A,set(A),set_ord_greaterThan(A),Yb) )
        <=> ( Xb = Yb ) ) ) ).

% greaterThan_eq_iff
tff(fact_5222_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( member(A,I2,aa(A,set(A),set_ord_greaterThan(A),K2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),I2) ) ) ).

% greaterThan_iff
tff(fact_5223_Inf__greaterThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_greaterThan(A),Xb)) = Xb ) ).

% Inf_greaterThan
tff(fact_5224_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),Xb)),aa(A,set(A),set_ord_greaterThan(A),Yb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ).

% greaterThan_subset_iff
tff(fact_5225_Compl__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K2: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_greaterThan(A),K2)) = aa(A,set(A),set_ord_atMost(A),K2) ) ).

% Compl_greaterThan
tff(fact_5226_Compl__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K2: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atMost(A),K2)) = aa(A,set(A),set_ord_greaterThan(A),K2) ) ).

% Compl_atMost
tff(fact_5227_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),top_top(A))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),Xb)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_5228_image__uminus__lessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_lessThan(A),Xb)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_lessThan
tff(fact_5229_image__uminus__greaterThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_greaterThan(A),Xb)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_greaterThan
tff(fact_5230_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_5231_infinite__Ioi,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A2: A] : ~ aa(set(A),$o,finite_finite2(A),aa(A,set(A),set_ord_greaterThan(A),A2)) ) ).

% infinite_Ioi
tff(fact_5232_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xb: A] : aa(A,set(A),set_ord_greaterThan(A),Xb) != bot_bot(set(A)) ) ).

% greaterThan_non_empty
tff(fact_5233_empty__in__Fpow,axiom,
    ! [A: $tType,A5: set(A)] : member(set(A),bot_bot(set(A)),finite_Fpow(A,A5)) ).

% empty_in_Fpow
tff(fact_5234_lessThan__Int__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: 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),A2)),aa(A,set(A),set_ord_greaterThan(A),B2)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% lessThan_Int_lessThan
tff(fact_5235_trivial__limit__at__right__real,axiom,
    ! [A: $tType] :
      ( ( dense_order(A)
        & no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A] : topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb)) != bot_bot(filter(A)) ) ).

% trivial_limit_at_right_real
tff(fact_5236_Fpow__not__empty,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) != bot_bot(set(set(A))) ).

% Fpow_not_empty
tff(fact_5237_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P2: fun(A,$o),Xb: A] :
          ( eventually(A,P2,topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb)))
        <=> ? [B7: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B7)
              & ! [Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),B7)
                   => aa(A,$o,P2,Y4) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_5238_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,Yb: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( eventually(A,P2,topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B7)
                & ! [Y4: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),B7)
                     => aa(A,$o,P2,Y4) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_5239_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_5240_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_5241_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_5242_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_5243_greaterThanLessThan__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] : set_or5935395276787703475ssThan(A,A2,B2) = 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),A2)),aa(A,set(A),set_ord_lessThan(A),B2)) ) ).

% greaterThanLessThan_eq
tff(fact_5244_greaterThanLessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or5935395276787703475ssThan(A,L,U) = 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),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).

% greaterThanLessThan_def
tff(fact_5245_greaterThanAtMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or3652927894154168847AtMost(A,L,U) = 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),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).

% greaterThanAtMost_def
tff(fact_5246_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),Xb),topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb))) ) ).

% eventually_at_right_less
tff(fact_5247_Fpow__mono,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A5)),finite_Fpow(A,B5)) ) ).

% Fpow_mono
tff(fact_5248_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ? [A4: A,B4: A] :
              ( member(A,Xb,aa(A,set(A),set_ord_lessThan(A),A4))
              & member(A,Yb,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)),aa(A,set(A),set_ord_lessThan(A),A4)),aa(A,set(A),set_ord_greaterThan(A),B4)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_5249_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P2: fun(A,$o)] :
          ( ! [X3: A] :
              ( member(A,X3,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P2,X3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P2,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% eventually_at_rightI
tff(fact_5250_Fpow__subset__Pow,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A5)),pow2(A,A5)) ).

% Fpow_subset_Pow
tff(fact_5251_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_5252_Fpow__def,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_wd(set(A),fun(set(A),$o),A5)) ).

% Fpow_def
tff(fact_5253_Fpow__Pow__finite,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A5)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_5254_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_ug(real,fun(real,fun(real,$o)),A2),B2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ).

% eventually_at_right_real
tff(fact_5255_greaterThan__Suc,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K2)) = 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),K2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,K2)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_5256_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),P: B,F12: filter(A),C3: B,L: B] :
          ( filterlim(A,B,F3,topolo174197925503356063within(B,P,aa(B,set(B),set_ord_greaterThan(B),P)),F12)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C3)
           => ( ( L = aa(B,B,aa(B,fun(B,B),times_times(B),C3),P) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_we(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo174197925503356063within(B,L,aa(B,set(B),set_ord_greaterThan(B),L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_5257_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(B,fun(A,$o)),F3),L5),F4)
           => filterlim(A,B,F3,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_greaterThan(B),L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_5258_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F3: fun(A,B),P2: fun(B,$o),G3: fun(B,A),A2: A] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P2,X3)
               => ( aa(A,B,F3,aa(B,A,G3,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P2,X3)
                 => aa(A,$o,Q,aa(B,A,G3,X3)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
               => ( ! [B4: A] :
                      ( aa(A,$o,Q,B4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B4) )
                 => ( eventually(B,P2,at_bot(B))
                   => filterlim(A,B,F3,at_bot(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_5259_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_5260_nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A] : topolo7230453075368039082e_nhds(A,Xb) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image2(real,filter(A),aTP_Lamp_wg(A,fun(real,filter(A)),Xb)),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% nhds_metric
tff(fact_5261_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_wh(A,fun(A,filter(A)),Xb)),aa(A,set(A),set_ord_greaterThan(A),Xb))) ) ) ) ).

% at_right_eq
tff(fact_5262_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,G3: fun(A,B),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2)),G3)
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,G3,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_wi(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G3),F3)) ) ) ) ).

% isCont_If_ge
tff(fact_5263_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A)] :
          ( ! [A4: A,B4: A,X3: A] :
              ( member(A,A4,S2)
             => ( member(A,B4,S2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B4)
                   => member(A,X3,S2) ) ) ) )
         => ? [A4: A,B4: A] :
              ( ( S2 = bot_bot(set(A)) )
              | ( S2 = top_top(set(A)) )
              | ( S2 = aa(A,set(A),set_ord_lessThan(A),B4) )
              | ( S2 = aa(A,set(A),set_ord_atMost(A),B4) )
              | ( S2 = aa(A,set(A),set_ord_greaterThan(A),A4) )
              | ( S2 = aa(A,set(A),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_5264_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: A,B2: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ! [F2: fun(nat,A)] :
                ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,F2,N4))
               => ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N4)),B2)
                 => ( order_antimono(nat,A,F2)
                   => ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_wj(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P2),F2),at_top(nat)) ) ) ) )
           => eventually(A,P2,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_5265_GMVT,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),G3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X3: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
       => ( ! [X3: real] :
              ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2) )
             => differentiable(real,real,F3,topolo174197925503356063within(real,X3,top_top(set(real)))) )
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),G3) )
           => ( ! [X3: real] :
                  ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2) )
                 => differentiable(real,real,G3,topolo174197925503356063within(real,X3,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_field_derivative(real,G3,G_c,topolo174197925503356063within(real,C2,top_top(set(real))))
                  & has_field_derivative(real,F3,F_c,topolo174197925503356063within(real,C2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),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,F3,B2)),aa(real,real,F3,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,G3,B2)),aa(real,real,G3,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_5266_atLeast__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,set(A),set_ord_atLeast(A),Xb) = aa(A,set(A),set_ord_atLeast(A),Yb) )
        <=> ( Xb = Yb ) ) ) ).

% atLeast_eq_iff
tff(fact_5267_atLeast__0,axiom,
    aa(nat,set(nat),set_ord_atLeast(nat),zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_5268_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K2: A] :
          ( member(A,I2,aa(A,set(A),set_ord_atLeast(A),K2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K2),I2) ) ) ).

% atLeast_iff
tff(fact_5269_Inf__atLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atLeast(A),Xb)) = Xb ) ).

% Inf_atLeast
tff(fact_5270_atLeast__empty__triv,axiom,
    ! [A: $tType] : aa(set(A),set(set(A)),set_ord_atLeast(set(A)),bot_bot(set(A))) = top_top(set(set(A))) ).

% atLeast_empty_triv
tff(fact_5271_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Yb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),Xb)),aa(A,set(A),set_ord_atLeast(A),Yb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb) ) ) ).

% atLeast_subset_iff
tff(fact_5272_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K2)),aa(A,set(A),set_ord_atLeast(A),I2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),I2)) ) ).

% image_add_atLeast
tff(fact_5273_Sup__atLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atLeast(A),Xb)) = top_top(A) ) ).

% Sup_atLeast
tff(fact_5274_Compl__atLeast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K2: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atLeast(A),K2)) = aa(A,set(A),set_ord_lessThan(A),K2) ) ).

% Compl_atLeast
tff(fact_5275_Compl__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K2: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_lessThan(A),K2)) = aa(A,set(A),set_ord_atLeast(A),K2) ) ).

% Compl_lessThan
tff(fact_5276_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atLeast(A),L2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L2),L) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_5277_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C3)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ).

% image_minus_const_AtMost
tff(fact_5278_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C3)),aa(A,set(A),set_ord_atLeast(A),A2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)) ) ).

% image_minus_const_atLeast
tff(fact_5279_image__uminus__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_atMost(A),Xb)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_atMost
tff(fact_5280_image__uminus__atLeast,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_atLeast(A),Xb)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_atLeast
tff(fact_5281_Int__atLeastAtMostR2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),A2)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C3),D3) ) ).

% Int_atLeastAtMostR2
tff(fact_5282_Int__atLeastAtMostL2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(A,set(A),set_ord_atLeast(A),C3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C3),B2) ) ).

% Int_atLeastAtMostL2
tff(fact_5283_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C3: B,Q2: fun(A,B),Ta: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wk(B,fun(fun(A,B),fun(A,B)),C3),Q2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C3 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_5284_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q2: fun(A,B),C3: B,Ta: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_wl(fun(A,B),fun(B,fun(A,B)),Q2),C3),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C3 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_5285_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K2)) = aa(nat,set(nat),set_ord_greaterThan(nat),K2) ).

% atLeast_Suc_greaterThan
tff(fact_5286_not__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atLeast(A),L) ) ).

% not_empty_eq_Ici_eq_empty
tff(fact_5287_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_5288_not__Ici__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L2: A,L: A,H: A] : aa(A,set(A),set_ord_atLeast(A),L2) != set_or1337092689740270186AtMost(A,L,H) ) ).

% not_Ici_eq_Icc
tff(fact_5289_not__Iic__eq__Ici,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A,L2: A] : aa(A,set(A),set_ord_atMost(A),H) != aa(A,set(A),set_ord_atLeast(A),L2) ) ).

% not_Iic_eq_Ici
tff(fact_5290_infinite__Ici,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A2: A] : ~ aa(set(A),$o,finite_finite2(A),aa(A,set(A),set_ord_atLeast(A),A2)) ) ).

% infinite_Ici
tff(fact_5291_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( differentiable(A,B,F3,F4)
         => ( differentiable(A,B,G3,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ph(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),F4) ) ) ) ).

% differentiable_add
tff(fact_5292_not__UNIV__eq__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L2: A] : top_top(set(A)) != aa(A,set(A),set_ord_atLeast(A),L2) ) ).

% not_UNIV_eq_Ici
tff(fact_5293_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_antimono(A,B,F3)
        <=> ! [X4: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Y4)),aa(A,B,F3,X4)) ) ) ) ).

% antimono_def
tff(fact_5294_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X3)) )
         => order_antimono(A,B,F3) ) ) ).

% antimonoI
tff(fact_5295_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_antimono(A,B,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Yb)),aa(A,B,F3,Xb)) ) ) ) ).

% antimonoE
tff(fact_5296_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_antimono(A,B,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Yb)),aa(A,B,F3,Xb)) ) ) ) ).

% antimonoD
tff(fact_5297_atLeast__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xb: A] :
          ( ( aa(A,set(A),set_ord_atLeast(A),Xb) = top_top(set(A)) )
        <=> ( Xb = bot_bot(A) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_5298_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atLeast(A),L)) ) ).

% not_UNIV_le_Ici
tff(fact_5299_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L2: A,H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),set_or1337092689740270186AtMost(A,L2,H2)) ) ).

% not_Ici_le_Icc
tff(fact_5300_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),H2)) ) ).

% not_Ici_le_Iic
tff(fact_5301_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),aa(A,set(A),set_ord_atLeast(A),L2)) ) ).

% not_Iic_le_Ici
tff(fact_5302_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),A2)),aa(A,set(A),set_ord_atLeast(A),A2)) ) ).

% Ioi_le_Ico
tff(fact_5303_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_antimono(nat,A,F3)
        <=> ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N))),aa(nat,A,F3,N)) ) ) ).

% decseq_Suc_iff
tff(fact_5304_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N2))),aa(nat,A,X5,N2))
         => order_antimono(nat,A,X5) ) ) ).

% decseq_SucI
tff(fact_5305_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: fun(nat,A),I2: nat] :
          ( order_antimono(nat,A,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A5,aa(nat,nat,suc,I2))),aa(nat,A,A5,I2)) ) ) ).

% decseq_SucD
tff(fact_5306_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( order_antimono(nat,A,X5)
        <=> ! [M2: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,M2)) ) ) ) ).

% decseq_def
tff(fact_5307_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),I2: nat,J: nat] :
          ( order_antimono(nat,A,F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,J)),aa(nat,A,F3,I2)) ) ) ) ).

% decseqD
tff(fact_5308_differentiable__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [S: set(A),F3: fun(A,fun(B,C)),Net: filter(B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => differentiable(B,C,aa(A,fun(B,C),F3,X3),Net) )
           => differentiable(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wn(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F3),Net) ) ) ) ).

% differentiable_sum
tff(fact_5309_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),A2)),aa(A,set(A),set_ord_greaterThan(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_5310_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)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(8)
tff(fact_5311_atLeastLessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or7035219750837199246ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).

% atLeastLessThan_def
tff(fact_5312_atLeastAtMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or1337092689740270186AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).

% atLeastAtMost_def
tff(fact_5313_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)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(6)
tff(fact_5314_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),Xb: A,S: set(A),G3: fun(A,B)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,Xb,S))
         => ( differentiable(A,B,G3,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,B,G3,Xb) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pk(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_5315_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),Xb: A,S: set(A)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,B,F3,Xb) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_wo(fun(A,B),fun(A,B),F3),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% differentiable_inverse
tff(fact_5316_atMost__Int__atLeast,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Nb: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),Nb)),aa(A,set(A),set_ord_atLeast(A),Nb)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Nb),bot_bot(set(A))) ) ).

% atMost_Int_atLeast
tff(fact_5317_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),L5: A,Nb: nat] :
          ( order_antimono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L5),aa(nat,A,X5,Nb)) ) ) ) ).

% decseq_ge
tff(fact_5318_UN__atLeast__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atLeast(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atLeast_UNIV
tff(fact_5319_atLeast__Suc,axiom,
    ! [K2: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atLeast(nat),K2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),K2),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_5320_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,X5: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ! [S6: fun(nat,A)] :
                ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,S6,N4))
               => ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S6,N4)),B2)
                 => ( order_antimono(nat,A,S6)
                   => ( filterlim(nat,A,S6,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wp(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S6),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_5321_continuous__at__Sup__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),S2: set(A)] :
          ( order_antimono(A,B,F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S2),aa(A,set(A),set_ord_lessThan(A),aa(set(A),A,complete_Sup_Sup(A),S2))),F3)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S2)
               => ( aa(A,B,F3,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,F3),S2)) ) ) ) ) ) ) ).

% continuous_at_Sup_antimono
tff(fact_5322_continuous__at__Inf__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),S2: set(A)] :
          ( order_antimono(A,B,F3)
         => ( 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))),F3)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S2)
               => ( aa(A,B,F3,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,F3),S2)) ) ) ) ) ) ) ).

% continuous_at_Inf_antimono
tff(fact_5323_MVT,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2)
               => differentiable(real,real,F3,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ? [L3: real,Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B2)
              & has_field_derivative(real,F3,L3,topolo174197925503356063within(real,Z2,top_top(set(real))))
              & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,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_5324_bdd__below_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: set(A),M4: A] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M4),X3) )
         => condit1013018076250108175_below(A,A5) ) ) ).

% bdd_below.I
tff(fact_5325_bdd__belowI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: set(A),Mb: A] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X3) )
         => condit1013018076250108175_below(A,A5) ) ) ).

% bdd_belowI
tff(fact_5326_bdd__above_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: set(A),M4: A] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),M4) )
         => condit941137186595557371_above(A,A5) ) ) ).

% bdd_above.I
tff(fact_5327_bdd__below__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit1013018076250108175_below(A,bot_bot(set(A))) ) ).

% bdd_below_empty
tff(fact_5328_bdd__above__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit941137186595557371_above(A,bot_bot(set(A))) ) ).

% bdd_above_empty
tff(fact_5329_bdd__below__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I6: set(A),A5: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( condit1013018076250108175_below(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I6)))
          <=> ! [X4: A] :
                ( member(A,X4,I6)
               => condit1013018076250108175_below(B,aa(A,set(B),A5,X4)) ) ) ) ) ).

% bdd_below_UN
tff(fact_5330_bdd__above__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I6: set(A),A5: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I6)))
          <=> ! [X4: A] :
                ( member(A,X4,I6)
               => condit941137186595557371_above(B,aa(A,set(B),A5,X4)) ) ) ) ) ).

% bdd_above_UN
tff(fact_5331_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A5)
           => ( condit1013018076250108175_below(A,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ) ).

% cInf_le_cSup
tff(fact_5332_continuous__on__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [S: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F3)
         => ( topolo81223032696312382ous_on(A,B,S,G3)
           => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_wq(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_on_add
tff(fact_5333_continuous__on__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A5,F3)
         => ( topolo81223032696312382ous_on(A,B,A5,G3)
           => topolo81223032696312382ous_on(A,B,A5,aa(fun(A,B),fun(A,B),aTP_Lamp_wr(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_on_max
tff(fact_5334_bdd__above__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => condit941137186595557371_above(A,A5) ) ) ).

% bdd_above_finite
tff(fact_5335_bdd__below__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => condit1013018076250108175_below(A,A5) ) ) ).

% bdd_below_finite
tff(fact_5336_bdd__above__nat,axiom,
    ! [X5: set(nat)] :
      ( condit941137186595557371_above(nat,X5)
    <=> aa(set(nat),$o,finite_finite2(nat),X5) ) ).

% bdd_above_nat
tff(fact_5337_continuous__on__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(A),F3: fun(A,B),G3: fun(A,C)] :
          ( topolo81223032696312382ous_on(A,B,S,F3)
         => ( topolo81223032696312382ous_on(A,C,S,G3)
           => topolo81223032696312382ous_on(A,product_prod(B,C),S,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ws(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% continuous_on_Pair
tff(fact_5338_continuous__on__sing,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,F3: fun(A,B)] : topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))),F3) ) ).

% continuous_on_sing
tff(fact_5339_IVT2_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F3: fun(B,A),B2: B,Yb: A,A2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,B2)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),aa(B,A,F3,A2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A2,B2),F3)
               => ? [X3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X3)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B2)
                    & ( aa(B,A,F3,X3) = Yb ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_5340_IVT_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F3: fun(B,A),A2: B,Yb: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,A2)),Yb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),aa(B,A,F3,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A2,B2),F3)
               => ? [X3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X3)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B2)
                    & ( aa(B,A,F3,X3) = Yb ) ) ) ) ) ) ) ).

% IVT'
tff(fact_5341_bdd__below_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: set(A)] :
          ( condit1013018076250108175_below(A,A5)
         => ~ ! [M7: A] :
                ~ ! [X: A] :
                    ( member(A,X,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M7),X) ) ) ) ).

% bdd_below.E
tff(fact_5342_bdd__below_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: set(A)] :
          ( condit1013018076250108175_below(A,A5)
        <=> ? [M8: A] :
            ! [X4: A] :
              ( member(A,X4,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M8),X4) ) ) ) ).

% bdd_below.unfold
tff(fact_5343_bdd__above_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: set(A)] :
          ( condit941137186595557371_above(A,A5)
         => ~ ! [M7: A] :
                ~ ! [X: A] :
                    ( member(A,X,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M7) ) ) ) ).

% bdd_above.E
tff(fact_5344_bdd__above_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: set(A)] :
          ( condit941137186595557371_above(A,A5)
        <=> ? [M8: A] :
            ! [X4: A] :
              ( member(A,X4,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M8) ) ) ) ).

% bdd_above.unfold
tff(fact_5345_continuous__on__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B)] : topolo81223032696312382ous_on(A,B,bot_bot(set(A)),F3) ) ).

% continuous_on_empty
tff(fact_5346_bdd__belowI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A5: set(A),Mb: B,F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F3,X3)) )
         => condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A5)) ) ) ).

% bdd_belowI2
tff(fact_5347_bdd__below_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A5: set(A),M4: B,F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M4),aa(A,B,F3,X3)) )
         => condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A5)) ) ) ).

% bdd_below.I2
tff(fact_5348_bdd__above_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A5: set(A),F3: fun(A,B),M4: B] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),M4) )
         => condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A5)) ) ) ).

% bdd_above.I2
tff(fact_5349_cInf__lower2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A),Yb: A] :
          ( member(A,Xb,X5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => ( condit1013018076250108175_below(A,X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Yb) ) ) ) ) ).

% cInf_lower2
tff(fact_5350_cInf__lower,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A)] :
          ( member(A,Xb,X5)
         => ( condit1013018076250108175_below(A,X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Xb) ) ) ) ).

% cInf_lower
tff(fact_5351_cSup__upper2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A),Yb: A] :
          ( member(A,Xb,X5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
           => ( condit941137186595557371_above(A,X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ).

% cSup_upper2
tff(fact_5352_cSup__upper,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A)] :
          ( member(A,Xb,X5)
         => ( condit941137186595557371_above(A,X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ).

% cSup_upper
tff(fact_5353_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F3)
         => ( topolo81223032696312382ous_on(A,B,S,G3)
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => ( aa(A,B,G3,X3) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_wt(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_on_divide
tff(fact_5354_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S: set(A),F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F3)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( aa(A,B,F3,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_wu(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_on_inverse
tff(fact_5355_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S: set(A),F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F3)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( aa(A,B,F3,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_wv(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_on_sgn
tff(fact_5356_continuous__onI__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & dense_order(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(B,A),A5: set(B)] :
          ( topolo1002775350975398744n_open(A,aa(set(B),set(A),image2(B,A,F3),A5))
         => ( ! [X3: B,Y: B] :
                ( member(B,X3,A5)
               => ( member(B,Y,A5)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y)) ) ) )
           => topolo81223032696312382ous_on(B,A,A5,F3) ) ) ) ).

% continuous_onI_mono
tff(fact_5357_cINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A5: set(B),Xb: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A5))
         => ( member(B,Xb,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(B,A,F3,Xb)) ) ) ) ).

% cINF_lower
tff(fact_5358_cINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A5: set(B),Xb: B,U: A] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A5))
         => ( member(B,Xb,A5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,Xb)),U)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5))),U) ) ) ) ) ).

% cINF_lower2
tff(fact_5359_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( ( B5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A5)
           => ( ! [B4: A] :
                  ( member(A,B4,B5)
                 => ? [X: A] :
                      ( member(A,X,A5)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B4) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ) ) ).

% cInf_mono
tff(fact_5360_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,complete_Inf_Inf(A),S2))
            <=> ! [X4: A] :
                  ( member(A,X4,S2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_5361_cSUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Xb: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,Xb,A5)
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ).

% cSUP_upper
tff(fact_5362_cSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A5: set(B),Xb: B,U: A] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A5))
         => ( member(B,Xb,A5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F3,Xb))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))) ) ) ) ) ).

% cSUP_upper2
tff(fact_5363_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Yb: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Yb)
            <=> ? [X4: A] :
                  ( member(A,X4,X5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Yb) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_5364_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( ( B5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A5)
           => ( ! [B4: A] :
                  ( member(A,B4,B5)
                 => ? [X: A] :
                      ( member(A,X,A5)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B4),X) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B5)),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ) ).

% cSup_mono
tff(fact_5365_cSup__le__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S2)),A2)
            <=> ! [X4: A] :
                  ( member(A,X4,S2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A2) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_5366_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Yb: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),aa(set(A),A,complete_Sup_Sup(A),X5))
            <=> ? [X4: A] :
                  ( member(A,X4,X5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),X4) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_5367_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F3)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G3)
           => topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ww(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3))) ) ) ) ).

% open_Collect_less
tff(fact_5368_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F3: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F3)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( cos(A,aa(A,A,F3,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_re(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_on_tan
tff(fact_5369_open__Collect__less__Int,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real),G3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ( topolo81223032696312382ous_on(A,real,S,G3)
           => ? [A6: set(A)] :
                ( topolo1002775350975398744n_open(A,A6)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),S) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_wx(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),S),F3),G3)) ) ) ) ) ) ).

% open_Collect_less_Int
tff(fact_5370_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F3: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F3)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( sin(A,aa(A,A,F3,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_qm(fun(A,A),fun(A,A),F3)) ) ) ) ).

% continuous_on_cot
tff(fact_5371_continuous__on__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A5: set(A),F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A5,F3)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => ( cosh(B,aa(A,B,F3,X3)) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,A5,aTP_Lamp_wy(fun(A,B),fun(A,B),F3)) ) ) ) ).

% continuous_on_tanh
tff(fact_5372_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A5: set(B),Yb: A,I2: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F3),A5))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),A5)))
           => ( member(B,I2,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),aa(B,A,F3,I2)) ) ) ) ) ).

% less_cINF_D
tff(fact_5373_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F3: fun(B,A),A5: set(B),Yb: A,I2: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F3),A5))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),A5))),Yb)
           => ( member(B,I2,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,I2)),Yb) ) ) ) ) ).

% cSUP_lessD
tff(fact_5374_cINF__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B5: set(A),F3: fun(C,B),A5: set(C),G3: fun(A,B)] :
          ( ( B5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F3),A5))
           => ( ! [M: A] :
                  ( member(A,M,B5)
                 => ? [X: C] :
                      ( member(C,X,A5)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F3,X)),aa(A,B,G3,M)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F3),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G3),B5))) ) ) ) ) ).

% cINF_mono
tff(fact_5375_le__cINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),U: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5)))
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,X4)) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_5376_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,B5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B5)),aa(set(A),A,complete_Inf_Inf(A),A5)) ) ) ) ) ).

% cInf_superset_mono
tff(fact_5377_cSUP__mono,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),G3: fun(C,B),B5: set(C),F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,G3),B5))
           => ( ! [N2: A] :
                  ( member(A,N2,A5)
                 => ? [X: C] :
                      ( member(C,X,B5)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,N2)),aa(C,B,G3,X)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,G3),B5))) ) ) ) ) ).

% cSUP_mono
tff(fact_5378_cSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),U: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),U)
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),U) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_5379_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ) ) ).

% cSup_subset_mono
tff(fact_5380_cInf__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( condit1013018076250108175_below(A,X5)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),X5)) = $ite(X5 = bot_bot(set(A)),A2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Inf_Inf(A),X5))) ) ) ) ).

% cInf_insert_If
tff(fact_5381_cInf__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X5)
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),X5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ) ).

% cInf_insert
tff(fact_5382_open__Collect__positive,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ? [A6: set(A)] :
              ( topolo1002775350975398744n_open(A,A6)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),S) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,real),fun(A,$o),aTP_Lamp_wz(set(A),fun(fun(A,real),fun(A,$o)),S),F3)) ) ) ) ) ).

% open_Collect_positive
tff(fact_5383_continuous__on__powr_H,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real),G3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ( topolo81223032696312382ous_on(A,real,S,G3)
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F3,X3))
                    & ( ( aa(A,real,F3,X3) = zero_zero(real) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,X3)) ) ) )
             => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_xa(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_5384_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real),G3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ( topolo81223032696312382ous_on(A,real,S,G3)
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,X3)) )
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ( aa(A,real,F3,X3) != one_one(real) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,S)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,X3)) )
                 => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_xb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_5385_cINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A5: set(A),F3: fun(A,B),A2: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),A2)
            <=> ? [X4: A] :
                  ( member(A,X4,A5)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),A2) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_5386_less__cSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A5: set(A),F3: fun(A,B),A2: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5)))
            <=> ? [X4: A] :
                  ( member(A,X4,A5)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(A,B,F3,X4)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_5387_cINF__inf__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G3),A5))
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G3),A5))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xc(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)),A5)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_5388_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,B2: A,F3: fun(A,A)] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B2)
               => ? [Y3: A] : has_field_derivative(A,F3,Y3,topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A2,B2),F3) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_5389_cINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A),F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G3),B5))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
             => ( ! [X3: A] :
                    ( member(A,X3,B5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G3,X3)),aa(A,B,F3,X3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G3),B5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_5390_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),F9: fun(real,fun(real,real))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ( aa(real,real,F3,A2) = aa(real,real,F3,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
         => ( ! [X3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2)
                 => has_derivative(real,real,F3,aa(real,fun(real,real),F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B2)
                & ! [X: real] : aa(real,real,aa(real,fun(real,real),F9,Z2),X) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_5391_cSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A),F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G3),B5))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
             => ( ! [X3: A] :
                    ( member(A,X3,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G3),B5))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_5392_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( condit1013018076250108175_below(A,A5)
         => ( condit1013018076250108175_below(A,B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))),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)),A5),B5))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_5393_cINF__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),A2: A] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,A2)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ) ).

% cINF_insert
tff(fact_5394_mvt,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),F9: fun(real,fun(real,real))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2)
               => has_derivative(real,real,F3,aa(real,fun(real,real),F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xi)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xi),B2)
                 => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,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_5395_cINF__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A5: set(A),B5: fun(A,set(B)),F3: fun(B,C)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => ( aa(A,set(B),B5,X3) != bot_bot(set(B)) ) )
           => ( condit1013018076250108175_below(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_xd(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B5),F3)),A5)))
             => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A5)))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xe(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B5),F3)),A5)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_5396_cSUP__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A5: set(A),B5: fun(A,set(B)),F3: fun(B,C)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => ( aa(A,set(B),B5,X3) != bot_bot(set(B)) ) )
           => ( condit941137186595557371_above(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_xd(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B5),F3)),A5)))
             => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A5)))) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xf(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B5),F3)),A5)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_5397_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,B2)),topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_5398_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_5399_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2)
               => has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( aa(real,real,F3,B2) = aa(real,real,F3,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_5400_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),zero_zero(real)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,B2)),aa(real,real,F3,A2)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_5401_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y3) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,A2)),aa(real,real,F3,B2)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_5402_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2)
               => has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xb)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),B2)
             => ( aa(real,real,F3,Xb) = aa(real,real,F3,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_5403_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A,B2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,B2)),topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2)))
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),B2)
                   => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,X3)),topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_5404_Rolle,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ( aa(real,real,F3,A2) = aa(real,real,F3,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
         => ( ! [X3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B2)
                 => differentiable(real,real,F3,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B2)
                & has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_5405_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_xh(real,filter(product_prod(A,A)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ).

% uniformity_dist
tff(fact_5406_compactE__image,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),C5: set(B),F3: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [T4: B] :
                ( member(B,T4,C5)
               => topolo1002775350975398744n_open(A,aa(B,set(A),F3,T4)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),C5)))
             => ~ ! [C7: set(B)] :
                    ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C7),C5)
                   => ( aa(set(B),$o,finite_finite2(B),C7)
                     => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),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),F3),C7))) ) ) ) ) ) ) ).

% compactE_image
tff(fact_5407_compactE,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),T7: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T7))
           => ( ! [B8: set(A)] :
                  ( member(set(A),B8,T7)
                 => topolo1002775350975398744n_open(A,B8) )
             => ~ ! [T8: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),T8),T7)
                   => ( aa(set(set(A)),$o,finite_finite2(set(A)),T8)
                     => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T8)) ) ) ) ) ) ) ).

% compactE
tff(fact_5408_uniformity__refl,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o),Xb: A] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Xb)) ) ) ).

% uniformity_refl
tff(fact_5409_uniformity__trans,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ? [D7: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),D7,topolo7806501430040627800ormity(A))
              & ! [X: A,Y3: A,Z3: A] :
                  ( aa(product_prod(A,A),$o,D7,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3))
                 => ( aa(product_prod(A,A),$o,D7,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3))
                   => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)) ) ) ) ) ) ).

% uniformity_trans
tff(fact_5410_uniformity__transE,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ~ ! [D7: fun(product_prod(A,A),$o)] :
                ( eventually(product_prod(A,A),D7,topolo7806501430040627800ormity(A))
               => ~ ! [X: A,Y3: A] :
                      ( aa(product_prod(A,A),$o,D7,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3))
                     => ! [Z3: A] :
                          ( aa(product_prod(A,A),$o,D7,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3))
                         => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)) ) ) ) ) ) ).

% uniformity_transE
tff(fact_5411_uniformity__bot,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ( topolo7806501430040627800ormity(A) != bot_bot(filter(product_prod(A,A))) ) ) ).

% uniformity_bot
tff(fact_5412_compact__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo2193935891317330818ompact(A,bot_bot(set(A))) ) ).

% compact_empty
tff(fact_5413_compact__attains__inf,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,S2)
                & ! [Xa: A] :
                    ( member(A,Xa,S2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa) ) ) ) ) ) ).

% compact_attains_inf
tff(fact_5414_compact__attains__sup,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,S2)
                & ! [Xa: A] :
                    ( member(A,Xa,S2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X3) ) ) ) ) ) ).

% compact_attains_sup
tff(fact_5415_uniformity__sym,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(A,A),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_xi(fun(product_prod(A,A),$o),fun(A,fun(A,$o)),E5)),topolo7806501430040627800ormity(A)) ) ) ).

% uniformity_sym
tff(fact_5416_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F3: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F3)
             => ? [X3: A] :
                  ( member(A,X3,S)
                  & ! [Xa: A] :
                      ( member(A,Xa,S)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xa)),aa(A,B,F3,X3)) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_5417_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F3: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F3)
             => ? [X3: A] :
                  ( member(A,X3,S)
                  & ! [Xa: A] :
                      ( member(A,Xa,S)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Xa)) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_5418_Cauchy__uniform__iff,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [P5: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),P5,topolo7806501430040627800ormity(A))
             => ? [N6: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                 => ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M2)
                     => aa(product_prod(A,A),$o,P5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,X5,N)),aa(nat,A,X5,M2))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_5419_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_xk(real,filter(product_prod(complex,complex)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_complex_def
tff(fact_5420_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_xm(real,filter(product_prod(real,real)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_real_def
tff(fact_5421_totally__bounded__def,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
        <=> ! [E6: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),E6,topolo7806501430040627800ormity(A))
             => ? [X9: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),X9)
                  & ! [X4: A] :
                      ( member(A,X4,S2)
                     => ? [Xa2: A] :
                          ( member(A,Xa2,X9)
                          & aa(product_prod(A,A),$o,E6,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4)) ) ) ) ) ) ) ).

% totally_bounded_def
tff(fact_5422_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P2: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),P2,topolo7806501430040627800ormity(A))
        <=> ? [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
              & ! [X4: A,Y4: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,Y4)),E4)
                 => aa(product_prod(A,A),$o,P2,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4)) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_5423_tendsto__iff__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E6: fun(product_prod(B,B),$o)] :
              ( eventually(product_prod(B,B),E6,topolo7806501430040627800ormity(B))
             => eventually(A,aa(fun(product_prod(B,B),$o),fun(A,$o),aa(B,fun(fun(product_prod(B,B),$o),fun(A,$o)),aTP_Lamp_xn(fun(A,B),fun(B,fun(fun(product_prod(B,B),$o),fun(A,$o))),F3),L),E6),F4) ) ) ) ).

% tendsto_iff_uniformity
tff(fact_5424_compact__eq__Heine__Borel,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
        <=> ! [C8: set(set(A))] :
              ( ( ! [X4: set(A)] :
                    ( member(set(A),X4,C8)
                   => topolo1002775350975398744n_open(A,X4) )
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C8)) )
             => ? [D8: set(set(A))] :
                  ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),D8),C8)
                  & aa(set(set(A)),$o,finite_finite2(set(A)),D8)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),D8)) ) ) ) ) ).

% compact_eq_Heine_Borel
tff(fact_5425_compactI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( ! [C6: set(set(A))] :
              ( ! [X: set(A)] :
                  ( member(set(A),X,C6)
                 => topolo1002775350975398744n_open(A,X) )
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6))
               => ? [C9: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C9),C6)
                    & aa(set(set(A)),$o,finite_finite2(set(A)),C9)
                    & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C9)) ) ) )
         => topolo2193935891317330818ompact(A,S) ) ) ).

% compactI
tff(fact_5426_prod__filter__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,I6: set(A),J4: set(B),A5: fun(A,filter(C)),B5: 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),A5),I6)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B5),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_xp(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J4),A5),B5)),I6)) ) ) ) ).

% prod_filter_INF
tff(fact_5427_prod__filter__INF1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I6: set(A),A5: fun(A,filter(B)),B5: 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),A5),I6)),B5) = 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_xq(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A5),B5)),I6)) ) ) ).

% prod_filter_INF1
tff(fact_5428_prod__filter__INF2,axiom,
    ! [C: $tType,B: $tType,A: $tType,J4: set(A),A5: filter(B),B5: fun(A,filter(C))] :
      ( ( J4 != bot_bot(set(A)) )
     => ( prod_filter(B,C,A5,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B5),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_xr(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A5),B5)),J4)) ) ) ).

% prod_filter_INF2
tff(fact_5429_prod__filter__eq__bot,axiom,
    ! [A: $tType,B: $tType,A5: filter(A),B5: filter(B)] :
      ( ( prod_filter(A,B,A5,B5) = bot_bot(filter(product_prod(A,B))) )
    <=> ( ( A5 = bot_bot(filter(A)) )
        | ( B5 = bot_bot(filter(B)) ) ) ) ).

% prod_filter_eq_bot
tff(fact_5430_eventually__prod__same,axiom,
    ! [A: $tType,P2: fun(product_prod(A,A),$o),F4: filter(A)] :
      ( eventually(product_prod(A,A),P2,prod_filter(A,A,F4,F4))
    <=> ? [Q7: fun(A,$o)] :
          ( eventually(A,Q7,F4)
          & ! [X4: A,Y4: A] :
              ( aa(A,$o,Q7,X4)
             => ( aa(A,$o,Q7,Y4)
               => aa(product_prod(A,A),$o,P2,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4)) ) ) ) ) ).

% eventually_prod_same
tff(fact_5431_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P2: fun(product_prod(A,B),$o),F4: filter(A),G5: filter(B)] :
      ( eventually(product_prod(A,B),P2,prod_filter(A,B,F4,G5))
    <=> ? [Pf: fun(A,$o),Pg: fun(B,$o)] :
          ( eventually(A,Pf,F4)
          & eventually(B,Pg,G5)
          & ! [X4: A,Y4: B] :
              ( aa(A,$o,Pf,X4)
             => ( aa(B,$o,Pg,Y4)
               => aa(product_prod(A,B),$o,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4)) ) ) ) ) ).

% eventually_prod_filter
tff(fact_5432_prod__filter__mono__iff,axiom,
    ! [A: $tType,B: $tType,A5: filter(A),B5: filter(B),C5: filter(A),D5: filter(B)] :
      ( ( A5 != bot_bot(filter(A)) )
     => ( ( B5 != bot_bot(filter(B)) )
       => ( aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,A5,B5)),prod_filter(A,B,C5,D5))
        <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),A5),C5)
            & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),B5),D5) ) ) ) ) ).

% prod_filter_mono_iff
tff(fact_5433_eventually__prod__sequentially,axiom,
    ! [P2: fun(product_prod(nat,nat),$o)] :
      ( eventually(product_prod(nat,nat),P2,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N6: nat] :
        ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M2)
         => ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
             => aa(product_prod(nat,nat),$o,P2,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N),M2)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_5434_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,A5: filter(A),P2: fun(B,$o),B5: filter(B)] :
      ( ( A5 != bot_bot(filter(A)) )
     => ( eventually(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_xs(fun(B,$o),fun(A,fun(B,$o)),P2)),prod_filter(A,B,A5,B5))
      <=> eventually(B,P2,B5) ) ) ).

% eventually_prod2
tff(fact_5435_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B5: filter(A),P2: fun(B,$o),A5: filter(B)] :
      ( ( B5 != bot_bot(filter(A)) )
     => ( eventually(product_prod(B,A),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),aTP_Lamp_xt(fun(B,$o),fun(B,fun(A,$o)),P2)),prod_filter(B,A,A5,B5))
      <=> eventually(B,P2,A5) ) ) ).

% eventually_prod1
tff(fact_5436_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_5437_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,B),G5: filter(B),F4: filter(A),G3: fun(A,C),H6: filter(C)] :
      ( filterlim(A,B,F3,G5,F4)
     => ( filterlim(A,C,G3,H6,F4)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_xu(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3),prod_filter(B,C,G5,H6),F4) ) ) ).

% filterlim_Pair
tff(fact_5438_tendsto__add__Pair,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [A2: A,B2: A] : filterlim(product_prod(A,A),A,aTP_Lamp_xv(product_prod(A,A),A),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),prod_filter(A,A,topolo7230453075368039082e_nhds(A,A2),topolo7230453075368039082e_nhds(A,B2))) ) ).

% tendsto_add_Pair
tff(fact_5439_uniformity__trans_H,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(product_prod(A,A),product_prod(A,A)),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_xx(fun(product_prod(A,A),$o),fun(A,fun(A,fun(product_prod(A,A),$o))),E5))),prod_filter(product_prod(A,A),product_prod(A,A),topolo7806501430040627800ormity(A),topolo7806501430040627800ormity(A))) ) ) ).

% uniformity_trans'
tff(fact_5440_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: B,Xb: A,S: set(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),topolo174197925503356063within(A,Xb,S))
        <=> ! [X9: fun(nat,A)] :
              ( ! [I4: nat] : member(A,aa(nat,A,X9,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)),insert(A),Xb),bot_bot(set(A)))))
             => ( filterlim(nat,A,X9,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
               => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),comp(A,B,nat,F3),X9),topolo7230453075368039082e_nhds(B,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_5441_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,A2: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( ! [F2: fun(nat,A)] :
                ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,F2,N4))
               => ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N4)),A2)
                 => ( order_mono(nat,A,F2)
                   => ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_wj(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P2),F2),at_top(nat)) ) ) ) )
           => eventually(A,P2,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_5442_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K2: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K2),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_xy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_xz(set(product_prod(A,A)),fun(nat,$o),R))))) ) ) ).

% relpow_finite_bounded1
tff(fact_5443_finite__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Nb: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ) ).

% finite_relpow
tff(fact_5444_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_mono(nat,A,aTP_Lamp_ya(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_5445_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),A5: A,B5: A,Nb: nat] :
          ( order_mono(A,A,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),A5)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),B5)) ) ) ) ).

% funpow_mono
tff(fact_5446_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_mono(A,B,F3)
        <=> ! [X4: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y4)) ) ) ) ).

% mono_def
tff(fact_5447_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) )
         => order_mono(A,B,F3) ) ) ).

% monoI
tff(fact_5448_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_mono(A,B,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb)) ) ) ) ).

% monoE
tff(fact_5449_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_mono(A,B,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb)) ) ) ) ).

% monoD
tff(fact_5450_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),I2: nat,J: nat] :
          ( order_mono(nat,A,F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,I2)),aa(nat,A,F3,J)) ) ) ) ).

% incseqD
tff(fact_5451_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( order_mono(nat,A,X5)
        <=> ! [M2: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M2)),aa(nat,A,X5,N)) ) ) ) ).

% incseq_def
tff(fact_5452_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: fun(nat,A),I2: nat] :
          ( order_mono(nat,A,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A5,I2)),aa(nat,A,A5,aa(nat,nat,suc,I2))) ) ) ).

% incseq_SucD
tff(fact_5453_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,aa(nat,nat,suc,N2)))
         => order_mono(nat,A,X5) ) ) ).

% incseq_SucI
tff(fact_5454_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_mono(nat,A,F3)
        <=> ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N))) ) ) ).

% incseq_Suc_iff
tff(fact_5455_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_mono(A,B,F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ) ).

% mono_invE
tff(fact_5456_relpow__Suc__D2_H,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A)),X: A,Y3: A,Z3: A] :
      ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
        & 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) )
     => ? [W: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),W),R)
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ) ).

% relpow_Suc_D2'
tff(fact_5457_comp__funpow,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F3: fun(B,B)] : aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aa(nat,fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),compow(fun(fun(A,B),fun(A,B))),Nb),comp(B,B,A,F3)) = comp(B,B,A,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Nb),F3)) ).

% comp_funpow
tff(fact_5458_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F3)
         => order_mono(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3)) ) ) ).

% mono_pow
tff(fact_5459_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_5460_mono__Suc,axiom,
    order_mono(nat,nat,suc) ).

% mono_Suc
tff(fact_5461_comp__cong,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F3: fun(B,A),G3: fun(C,B),Xb: C,F9: fun(D,A),G6: fun(E,D),X6: E] :
      ( ( aa(B,A,F3,aa(C,B,G3,Xb)) = aa(D,A,F9,aa(E,D,G6,X6)) )
     => ( aa(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G3),Xb) = aa(E,A,aa(fun(E,D),fun(E,A),comp(D,A,E,F9),G6),X6) ) ) ).

% comp_cong
tff(fact_5462_max__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),Mb: A,Nb: A] :
          ( order_mono(A,B,F3)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F3,Mb)),aa(A,B,F3,Nb)) = aa(A,B,F3,aa(A,A,aa(A,fun(A,A),ord_max(A),Mb),Nb)) ) ) ) ).

% max_of_mono
tff(fact_5463_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A2)) ) ).

% mono_add
tff(fact_5464_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_mono(A,B,F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb) ) ) ) ).

% mono_strict_invE
tff(fact_5465_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F3: fun(A,B),A5: A,B5: A] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B5))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,A5)),aa(A,B,F3,B5))) ) ) ).

% mono_inf
tff(fact_5466_funpow__Suc__right,axiom,
    ! [A: $tType,Nb: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3)),F3) ).

% funpow_Suc_right
tff(fact_5467_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3)) ).

% funpow.simps(2)
tff(fact_5468_funpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F3)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3)) ).

% funpow_add
tff(fact_5469_relpow__Suc__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R))
     => ~ ! [Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z),R) ) ) ).

% relpow_Suc_E
tff(fact_5470_relpow__Suc__I,axiom,
    ! [A: $tType,Xb: A,Yb: A,Nb: nat,R: set(product_prod(A,A)),Z: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Yb),Z),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R)) ) ) ).

% relpow_Suc_I
tff(fact_5471_relpow__Suc__D2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R))
     => ? [Y: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R)
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ) ).

% relpow_Suc_D2
tff(fact_5472_relpow__Suc__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R))
     => ~ ! [Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R)
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ) ).

% relpow_Suc_E2
tff(fact_5473_relpow__Suc__I2,axiom,
    ! [A: $tType,Xb: A,Yb: A,R: set(product_prod(A,A)),Z: A,Nb: nat] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Yb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R)) ) ) ).

% relpow_Suc_I2
tff(fact_5474_relpow__0__E,axiom,
    ! [A: $tType,Xb: A,Yb: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),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))
     => ( Xb = Yb ) ) ).

% relpow_0_E
tff(fact_5475_relpow__0__I,axiom,
    ! [A: $tType,Xb: A,R: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Xb),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R)) ).

% relpow_0_I
tff(fact_5476_cclfp__lowerbound,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F3: fun(A,A),A5: A] :
          ( order_mono(A,A,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F3,A5)),A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),order_532582986084564980_cclfp(A,F3)),A5) ) ) ) ).

% cclfp_lowerbound
tff(fact_5477_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => order_mono(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb)) ) ).

% mono_times_nat
tff(fact_5478_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => order_mono(A,A,aa(A,fun(A,A),times_times(A),A2)) ) ) ).

% mono_mult
tff(fact_5479_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F3: fun(A,B),Mb: A,Nb: A,M6: B,N5: B] :
          ( order_mono(A,B,F3)
         => ( ( aa(set(A),set(B),image2(A,B,F3),set_or7035219750837199246ssThan(A,Mb,Nb)) = set_or7035219750837199246ssThan(B,M6,N5) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb)
             => ( aa(A,B,F3,Mb) = M6 ) ) ) ) ) ).

% mono_image_least
tff(fact_5480_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G3: fun(C,B),A5: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X3: B,Y: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X3)),aa(B,A,H,Y))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,H),G3)),A5) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),A5)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_5481_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F3: fun(A,A),P: A,K2: nat] :
          ( order_mono(A,A,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P),aa(A,A,F3,P))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),top_top(A))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_5482_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F3: fun(A,A),P: A,K2: nat] :
          ( order_mono(A,A,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F3,P)),P)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),bot_bot(A))),P) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_5483_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),I2: nat,J: nat,Xb: A,Yb: A] :
          ( order_mono(A,A,F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,F3,Xb))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I2),F3),Xb)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J),F3),Yb)) ) ) ) ) ) ).

% funpow_mono2
tff(fact_5484_relpowp__relpow__eq,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A)),X: A,Xa: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),aTP_Lamp_yb(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X),Xa)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ) ).

% relpowp_relpow_eq
tff(fact_5485_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = 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,G3),suc)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_5486_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = 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,G3),suc)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_5487_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = 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,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_5488_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = 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,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_5489_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_5490_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_5491_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = 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,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_5492_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = 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,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_5493_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(fun(nat,nat),fun(nat,$o),comp(nat,$o,nat,bit_se5641148757651400278ts_bit(A,A2)),aa(nat,fun(nat,nat),plus_plus(nat),Nb)) ) ).

% bit_drop_bit_eq
tff(fact_5494_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ).

% mono_Sup
tff(fact_5495_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: fun(C,A),I6: set(C)] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yc(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5)),I6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A5),I6)))) ) ) ).

% mono_SUP
tff(fact_5496_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ).

% mono_Inf
tff(fact_5497_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F3: fun(A,B),A5: fun(C,A),I6: set(C)] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A5),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_yc(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5)),I6))) ) ) ).

% mono_INF
tff(fact_5498_relpow__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R)
               => ~ member(product_prod(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))),M),R)) ) ) ) ) ).

% relpow_E2
tff(fact_5499_relpow__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z),R) ) ) ) ) ).

% relpow_E
tff(fact_5500_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_antimono(nat,A,aTP_Lamp_yd(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_5501_relpow__empty,axiom,
    ! [A: $tType,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_5502_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),L5: A,Nb: nat] :
          ( order_mono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,Nb)),L5) ) ) ) ).

% incseq_le
tff(fact_5503_sum_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A5: set(A),H: fun(A,B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X3: A,Y: A] :
                ( member(A,X3,A5)
               => ( member(A,Y,A5)
                 => ( ( X3 != Y )
                   => ( ( aa(A,B,H,X3) = aa(A,B,H,Y) )
                     => ( aa(B,C,G3,aa(A,B,H,X3)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G3),aa(set(A),set(B),image2(A,B,H),A5)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G3),H)),A5) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_5504_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Mb: nat,Nb: nat,F3: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( order_mono(A,A,F3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F3),top_top(A))) ) ) ) ).

% funpow_increasing
tff(fact_5505_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Mb: nat,Nb: nat,F3: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( order_mono(A,A,F3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F3),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_5506_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A5: set(A),H: fun(A,B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X3: A,Y: A] :
                ( member(A,X3,A5)
               => ( member(A,Y,A5)
                 => ( ( X3 != Y )
                   => ( ( aa(A,B,H,X3) = aa(A,B,H,Y) )
                     => ( aa(B,C,G3,aa(A,B,H,X3)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G3),aa(set(A),set(B),image2(A,B,H),A5)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G3),H)),A5) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_5507_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Yb: A,Z: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,F3,Yb,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S2)))
         => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(A,fun(A,A),plus_plus(A),Z)),Yb,topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_5508_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(A,B,F3,aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F3),A5)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_5509_relpow__fun__conv,axiom,
    ! [A: $tType,A2: A,B2: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = A2 )
          & ( aa(nat,A,F7,Nb) = B2 )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
             => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F7,I4)),aa(nat,A,F7,aa(nat,nat,suc,I4))),R) ) ) ) ).

% relpow_fun_conv
tff(fact_5510_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),G3: fun(C,B),F3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ! [I3: A] :
                ( member(A,I3,I6)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G3,aa(A,C,F3,I3))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),aa(set(A),set(C),image2(A,C,F3),I6))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,C),fun(A,B),comp(C,B,A,G3),F3)),I6)) ) ) ) ).

% sum_image_le
tff(fact_5511_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),D3: A,F4: filter(B),A2: A] :
          ( filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F3),aa(A,fun(A,A),plus_plus(A),D3)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),top_top(set(A))))
        <=> filterlim(A,B,F3,F4,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_5512_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),F4: filter(B),A2: A,D3: A] :
          ( filterlim(A,B,F3,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F3),aa(A,fun(A,A),plus_plus(A),D3)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_5513_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,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,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_5514_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,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,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_5515_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,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,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_5516_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,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,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_5517_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) = 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,G3),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_5518_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_5519_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A5: fun(C,A),I6: set(C)] :
          ( order_mono(A,B,F3)
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,A5),I6))
           => ( ( I6 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ye(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5)),I6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A5),I6)))) ) ) ) ) ).

% mono_cSUP
tff(fact_5520_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => ( condit941137186595557371_above(A,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ) ).

% mono_cSup
tff(fact_5521_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F3: fun(A,B),A5: fun(C,A),I6: set(C)] :
          ( order_mono(A,B,F3)
         => ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,A5),I6))
           => ( ( I6 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A5),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_ye(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5)),I6))) ) ) ) ) ).

% mono_cINF
tff(fact_5522_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => ( condit1013018076250108175_below(A,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ) ).

% mono_cInf
tff(fact_5523_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: 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,G3),aTP_Lamp_yf(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_5524_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: 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,G3),aTP_Lamp_yf(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_5525_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: 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,G3),aTP_Lamp_yf(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_5526_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: 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,G3),aTP_Lamp_yf(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_5527_relpow__finite__bounded,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K2: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K2),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_xy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_yg(set(product_prod(A,A)),fun(nat,$o),R))))) ) ).

% relpow_finite_bounded
tff(fact_5528_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(int,A),Mb: nat,Nb: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G3),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Mb),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_5529_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)
     => order_mono(nat,nat,aTP_Lamp_yh(nat,fun(nat,nat),K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_5530_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(int,A),Mb: nat,Nb: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G3),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Mb),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_5531_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(int,A),Mb: nat,Nb: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G3),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),Mb),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_5532_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = 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,G3),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_5533_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_5534_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(int,A),Mb: nat,Nb: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G3),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),Mb),aa(nat,int,semiring_1_of_nat(int),Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),comp(int,A,nat,G3),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_5535_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F3),top_top(set(nat))))
         => ( order_mono(nat,A,F3)
           => ( ! [N2: nat] :
                  ( ( aa(nat,A,F3,N2) = aa(nat,A,F3,aa(nat,nat,suc,N2)) )
                 => ( aa(nat,A,F3,aa(nat,nat,suc,N2)) = aa(nat,A,F3,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) )
             => ? [N8: nat] :
                  ( ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N8)
                     => ! [M3: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N8)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N4)
                           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,M3)),aa(nat,A,F3,N4)) ) ) )
                  & ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N4)
                     => ( aa(nat,A,F3,N8) = aa(nat,A,F3,N4) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_5536_tendsto__at__left__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [B2: A,A2: A,X5: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( ! [S6: fun(nat,A)] :
                ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S6,N4)),A2)
               => ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,S6,N4))
                 => ( order_mono(nat,A,S6)
                   => ( filterlim(nat,A,S6,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wp(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S6),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_5537_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_5538_continuous__at__Sup__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),S2: set(A)] :
          ( order_mono(A,B,F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S2),aa(A,set(A),set_ord_lessThan(A),aa(set(A),A,complete_Sup_Sup(A),S2))),F3)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S2)
               => ( aa(A,B,F3,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,F3),S2)) ) ) ) ) ) ) ).

% continuous_at_Sup_mono
tff(fact_5539_continuous__at__Inf__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),S2: set(A)] :
          ( order_mono(A,B,F3)
         => ( 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))),F3)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S2)
               => ( aa(A,B,F3,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,F3),S2)) ) ) ) ) ) ) ).

% continuous_at_Inf_mono
tff(fact_5540_ntrancl__def,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,Nb,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_yi(nat,fun(nat,$o),Nb)))) ).

% ntrancl_def
tff(fact_5541_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( transitive_trancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_xz(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_5542_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F7: fun(nat,nat)] :
          ( order_mono(nat,nat,F7)
          & ( aa(set(nat),set(nat),image2(nat,nat,F7),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
             => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),aa(nat,nat,F7,I4)) ) )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) )
              <=> ( aa(nat,nat,F7,I4) = aa(nat,nat,F7,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_5543_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_5544_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_5545_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_5546_finite__trancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),transitive_trancl(A,R2))
    <=> aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2) ) ).

% finite_trancl
tff(fact_5547_ntrancl__Zero,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R) = R ).

% ntrancl_Zero
tff(fact_5548_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_yj(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_5549_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_yk(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_5550_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_yj(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_5551_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_yk(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_5552_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Yb: A,Xb: 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_mr(A,A)),Yb)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Xb)) = 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_mr(A,A)),Xb)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Yb)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_5553_remdups__adj__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups_adj(A,Xs) = Xs ) ) ).

% remdups_adj_distinct
tff(fact_5554_remdups__adj_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups_adj(A,nil(A)) = nil(A) ).

% remdups_adj.simps(1)
tff(fact_5555_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),R2)
       => ~ ! [B4: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4),transitive_trancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22),R2) ) ) ) ).

% trancl.cases
tff(fact_5556_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R2))
    <=> ( ? [A7: A,B7: A] :
            ( ( A1 = A7 )
            & ( A22 = B7 )
            & 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,C4: A] :
            ( ( A1 = A7 )
            & ( A22 = C4 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),transitive_trancl(A,R2))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C4),R2) ) ) ) ).

% trancl.simps
tff(fact_5557_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2)) ) ).

% trancl.r_into_trancl
tff(fact_5558_tranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
       => ~ ! [C2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2),transitive_trancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C2),B2),R2) ) ) ) ).

% tranclE
tff(fact_5559_trancl__trans,axiom,
    ! [A: $tType,Xb: A,Yb: A,R2: set(product_prod(A,A)),Z: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Yb),Z),transitive_trancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),transitive_trancl(A,R2)) ) ) ).

% trancl_trans
tff(fact_5560_trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P2: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( ! [Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y),R2)
           => aa(A,$o,P2,Y) )
       => ( ! [Y: A,Z2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y),transitive_trancl(A,R2))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
               => ( aa(A,$o,P2,Y)
                 => aa(A,$o,P2,Z2) ) ) )
         => aa(A,$o,P2,B2) ) ) ) ).

% trancl_induct
tff(fact_5561_r__r__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R)) ) ) ).

% r_r_into_trancl
tff(fact_5562_converse__tranclE,axiom,
    ! [A: $tType,Xb: A,Z: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),R2)
       => ~ ! [Y: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R2)
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z),transitive_trancl(A,R2)) ) ) ) ).

% converse_tranclE
tff(fact_5563_irrefl__trancl__rD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: A,Yb: A] :
      ( ! [X3: A] : ~ 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))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
       => ( Xb != Yb ) ) ) ).

% irrefl_trancl_rD
tff(fact_5564_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R2)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_5565_trancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_trancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R2)) ) ) ).

% trancl_into_trancl2
tff(fact_5566_trancl__trans__induct,axiom,
    ! [A: $tType,Xb: A,Yb: A,R2: set(product_prod(A,A)),P2: fun(A,fun(A,$o))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_trancl(A,R2))
     => ( ! [X3: A,Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),R2)
           => aa(A,$o,aa(A,fun(A,$o),P2,X3),Y) )
       => ( ! [X3: A,Y: A,Z2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),transitive_trancl(A,R2))
             => ( aa(A,$o,aa(A,fun(A,$o),P2,X3),Y)
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R2))
                 => ( aa(A,$o,aa(A,fun(A,$o),P2,Y),Z2)
                   => aa(A,$o,aa(A,fun(A,$o),P2,X3),Z2) ) ) ) )
         => aa(A,$o,aa(A,fun(A,$o),P2,Xb),Yb) ) ) ) ).

% trancl_trans_induct
tff(fact_5567_converse__trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P2: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( ! [Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),B2),R2)
           => aa(A,$o,P2,Y) )
       => ( ! [Y: A,Z2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),B2),transitive_trancl(A,R2))
               => ( aa(A,$o,P2,Z2)
                 => aa(A,$o,P2,Y) ) ) )
         => aa(A,$o,P2,A2) ) ) ) ).

% converse_trancl_induct
tff(fact_5568_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_trancl(product_prod(A,B),R2))
     => ( ! [A4: A,B4: B] :
            ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),R2)
           => aa(B,$o,aa(A,fun(B,$o),P2,A4),B4) )
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),transitive_trancl(product_prod(A,B),R2))
             => ( 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)
               => ( aa(B,$o,aa(A,fun(B,$o),P2,A4),B4)
                 => aa(B,$o,aa(A,fun(B,$o),P2,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P2,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_5569_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% remdups_adj_length
tff(fact_5570_finite__trancl__ntranl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( transitive_trancl(A,R) = 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_5571_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_5572_trancl__power,axiom,
    ! [A: $tType,P: product_prod(A,A),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P,transitive_trancl(A,R))
    <=> ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
          & member(product_prod(A,A),P,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R)) ) ) ).

% trancl_power
tff(fact_5573_fst__diag__id,axiom,
    ! [A: $tType,Z: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_fst(A,A)),aTP_Lamp_yk(A,product_prod(A,A))),Z) = aa(A,A,id(A),Z) ).

% fst_diag_id
tff(fact_5574_snd__diag__id,axiom,
    ! [A: $tType,Z: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_snd(A,A)),aTP_Lamp_yk(A,product_prod(A,A))),Z) = aa(A,A,id(A),Z) ).

% snd_diag_id
tff(fact_5575_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(D,fun(C,A)),G3: fun(B,D),Xb: product_prod(B,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),aa(fun(B,D),fun(B,fun(C,A)),comp(D,fun(C,A),B,F3),G3)),Xb) = aa(C,A,aa(D,fun(C,A),F3,aa(B,D,G3,aa(product_prod(B,C),B,product_fst(B,C),Xb))),aa(product_prod(B,C),C,product_snd(B,C),Xb)) ).

% case_prod_comp
tff(fact_5576_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(set(A)),G3: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,B5)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [A13: set(A)] :
                ( member(set(A),A13,B5)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B5)
                   => ( ( A13 != A24 )
                     => ! [X3: A] :
                          ( member(A,X3,A13)
                         => ( member(A,X3,A24)
                           => ( aa(A,B,G3,X3) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G3),B5) ) ) ) ) ).

% prod.Union_comp
tff(fact_5577_remdups__adj__adjacent,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I2) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I2)) ) ) ).

% remdups_adj_adjacent
tff(fact_5578_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S2)
    <=> ~ aa(set(nat),$o,finite_finite2(nat),aa(set(int),set(nat),image2(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),S2)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_5579_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_yl(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_5580_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_ym(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_5581_prod_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(set(A)),G3: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,C5)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [X3: set(A)] :
                ( member(set(A),X3,C5)
               => ! [Xa3: set(A)] :
                    ( member(set(A),Xa3,C5)
                   => ( ( X3 != Xa3 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa3) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G3),C5) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_5582_sum_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B5: set(set(A)),G3: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,B5)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [A13: set(A)] :
                ( member(set(A),A13,B5)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B5)
                   => ( ( A13 != A24 )
                     => ! [X3: A] :
                          ( member(A,X3,A13)
                         => ( member(A,X3,A24)
                           => ( aa(A,B,G3,X3) = zero_zero(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G3),B5) ) ) ) ) ).

% sum.Union_comp
tff(fact_5583_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))) ) ).

% remdups_adj_length_ge1
tff(fact_5584_sum_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(set(A)),G3: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,C5)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [X3: set(A)] :
                ( member(set(A),X3,C5)
               => ! [Xa3: set(A)] :
                    ( member(set(A),Xa3,C5)
                   => ( ( X3 != Xa3 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa3) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G3),C5) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_5585_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_uo(fun(A,real),fun(A,$o),F3),F4)
     => ( filterlim(A,real,F3,at_top(real),F4)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_5586_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(A,C),G3: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_yn(C,set(B))),F3) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image2(D,B,G3)),aTP_Lamp_yo(A,set(D))) ).

% empty_natural
tff(fact_5587_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: filter(A),G5: filter(B),H6: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F4,G5),H6) = filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_yq(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,H6))) ).

% prod_filter_assoc
tff(fact_5588_range__abs__Nats,axiom,
    aa(set(int),set(int),image2(int,int,abs_abs(int)),top_top(set(int))) = semiring_1_Nats(int) ).

% range_abs_Nats
tff(fact_5589_filtermap__bot,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] : filtermap(B,A,F3,bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtermap_bot
tff(fact_5590_filtermap__bot__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),F4: filter(B)] :
      ( ( filtermap(B,A,F3,F4) = bot_bot(filter(A)) )
    <=> ( F4 = bot_bot(filter(B)) ) ) ).

% filtermap_bot_iff
tff(fact_5591_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Nb),semiring_1_Nats(A)) ) ).

% of_nat_in_Nats
tff(fact_5592_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: A,P2: fun(A,$o)] :
          ( member(A,Xb,semiring_1_Nats(A))
         => ( ! [N2: nat] : aa(A,$o,P2,aa(nat,A,semiring_1_of_nat(A),N2))
           => aa(A,$o,P2,Xb) ) ) ) ).

% Nats_induct
tff(fact_5593_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: A] :
          ( member(A,Xb,semiring_1_Nats(A))
         => ~ ! [N2: nat] : Xb != aa(nat,A,semiring_1_of_nat(A),N2) ) ) ).

% Nats_cases
tff(fact_5594_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,zero_zero(A),semiring_1_Nats(A)) ) ).

% Nats_0
tff(fact_5595_Nats__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [W2: num] : member(A,aa(num,A,numeral_numeral(A),W2),semiring_1_Nats(A)) ) ).

% Nats_numeral
tff(fact_5596_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),semiring_1_Nats(A)) ) ) ) ).

% Nats_mult
tff(fact_5597_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,one_one(A),semiring_1_Nats(A)) ) ).

% Nats_1
tff(fact_5598_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),semiring_1_Nats(A)) ) ) ) ).

% Nats_add
tff(fact_5599_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),G3: fun(C,B),F4: filter(C)] : aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_yr(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F3),G3),F4)),prod_filter(A,B,filtermap(C,A,F3,F4),filtermap(C,B,G3,F4))) ).

% filtermap_Pair
tff(fact_5600_filtermap__nhds__times,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C3),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)) ) ) ) ).

% filtermap_nhds_times
tff(fact_5601_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,semiring_1_Nats(A))
         => ( member(A,B2,semiring_1_Nats(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),semiring_1_Nats(A)) ) ) ) ) ).

% Nats_diff
tff(fact_5602_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,top_top(set(A))) = filtermap(A,A,aTP_Lamp_ys(A,fun(A,A),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% at_to_0
tff(fact_5603_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = aa(set(nat),set(A),image2(nat,A,semiring_1_of_nat(A)),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_5604_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C3: A,P: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C3),topolo174197925503356063within(A,P,aa(A,set(A),set_ord_greaterThan(A),P))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),P),aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),P))) ) ) ) ).

% filtermap_times_pos_at_right
tff(fact_5605_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_5606_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,Xb: A,F4: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))),F4) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),F4) ).

% prod_filter_principal_singleton
tff(fact_5607_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),Xb: B] : prod_filter(A,B,F4,principal(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ym(B,fun(A,product_prod(A,B))),Xb),F4) ).

% prod_filter_principal_singleton2
tff(fact_5608_cauchy__filter__metric__filtermap,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F3: fun(B,A),F4: filter(B)] :
          ( topolo6773858410816713723filter(A,filtermap(B,A,F3,F4))
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [P5: fun(B,$o)] :
                  ( eventually(B,P5,F4)
                  & ! [X4: B,Y4: B] :
                      ( ( aa(B,$o,P5,X4)
                        & aa(B,$o,P5,Y4) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(B,A,F3,X4),aa(B,A,F3,Y4))),E4) ) ) ) ) ) ).

% cauchy_filter_metric_filtermap
tff(fact_5609_surj__int__encode,axiom,
    aa(set(int),set(nat),image2(int,nat,nat_int_encode),top_top(set(int))) = top_top(set(nat)) ).

% surj_int_encode
tff(fact_5610_compact__imp__fip__image,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),I6: set(B),F3: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [I3: B] :
                ( member(B,I3,I6)
               => topolo7761053866217962861closed(A,aa(B,set(A),F3,I3)) )
           => ( ! [I7: set(B)] :
                  ( aa(set(B),$o,finite_finite2(B),I7)
                 => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),I7),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),F3),I7))) != 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),F3),I6))) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip_image
tff(fact_5611_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => inj_on(real,real,aTP_Lamp_nq(nat,fun(real,real),Nb),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_5612_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : inj_on(A,B,F3,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_5613_closed__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo7761053866217962861closed(A,bot_bot(set(A))) ) ).

% closed_empty
tff(fact_5614_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)),insert(A),A2),bot_bot(set(A)))) ) ).

% closed_singleton
tff(fact_5615_closed__Union,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(set(A))] :
          ( aa(set(set(A)),$o,finite_finite2(set(A)),S2)
         => ( ! [X3: set(A)] :
                ( member(set(A),X3,S2)
               => topolo7761053866217962861closed(A,X3) )
           => topolo7761053866217962861closed(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S2)) ) ) ) ).

% closed_Union
tff(fact_5616_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_5617_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_yt(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_5618_closed__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A5: set(A),B5: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => topolo7761053866217962861closed(B,aa(A,set(B),B5,X3)) )
           => topolo7761053866217962861closed(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B5),A5))) ) ) ) ).

% closed_UN
tff(fact_5619_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A2: A,A5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5))
    <=> ( inj_on(A,B,F3,A5)
        & ~ member(B,aa(A,B,F3,A2),aa(set(A),set(B),image2(A,B,F3),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)),insert(A),A2),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_5620_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(A),A5: set(A)] :
      ( inj_on(A,B,F3,B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),aa(set(A),set(B),image2(A,B,F3),B5)) ) ) ).

% inj_on_strict_subset
tff(fact_5621_inj__on__image__Fpow,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => inj_on(set(A),set(B),image2(A,B,F3),finite_Fpow(A,A5)) ) ).

% inj_on_image_Fpow
tff(fact_5622_finite__image__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,F3),A5))
      <=> aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% finite_image_iff
tff(fact_5623_finite__imageD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5))
     => ( inj_on(B,A,F3,A5)
       => aa(set(B),$o,finite_finite2(B),A5) ) ) ).

% finite_imageD
tff(fact_5624_card__image,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F3),A5)) = aa(set(A),nat,finite_card(A),A5) ) ) ).

% card_image
tff(fact_5625_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
             => ( member(A,X3,A5)
               => ( member(A,Y,A5)
                 => ( aa(A,B,F3,X3) != aa(A,B,F3,Y) ) ) ) )
         => ( ! [X3: A,Y: A] :
                ( member(A,X3,A5)
               => ( member(A,Y,A5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X3) ) ) )
           => inj_on(A,B,F3,A5) ) ) ) ).

% linorder_inj_onI
tff(fact_5626_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),top_top(set(A))) ) ).

% inj_add_left
tff(fact_5627_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_mr(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_5628_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A5: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),A5) ) ).

% inj_on_add
tff(fact_5629_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A5: set(A)] : inj_on(A,A,aTP_Lamp_yu(A,fun(A,A),A2),A5) ) ).

% inj_on_add'
tff(fact_5630_int__encode__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa(int,nat,nat_int_encode,Xb) = aa(int,nat,nat_int_encode,Yb) )
    <=> ( Xb = Yb ) ) ).

% int_encode_eq
tff(fact_5631_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(B,A),D5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( inj_on(B,A,F3,D5)
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_yv(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A5),F3),D5))) ) ) ).

% finite_inverse_image_gen
tff(fact_5632_finite__imp__closed,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => topolo7761053866217962861closed(A,S2) ) ) ).

% finite_imp_closed
tff(fact_5633_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A5: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A5) ) ) ).

% inj_on_mult
tff(fact_5634_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
             => ( aa(A,B,F3,X3) != aa(A,B,F3,Y) ) )
         => inj_on(A,B,F3,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_5635_inj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),Nb: nat] :
      ( inj_on(A,A,F3,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),top_top(set(A))) ) ).

% inj_fn
tff(fact_5636_inj__on__Inter,axiom,
    ! [B: $tType,A: $tType,S2: set(set(A)),F3: fun(A,B)] :
      ( ( S2 != bot_bot(set(set(A))) )
     => ( ! [A6: set(A)] :
            ( member(set(A),A6,S2)
           => inj_on(A,B,F3,A6) )
       => inj_on(A,B,F3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S2)) ) ) ).

% inj_on_Inter
tff(fact_5637_filtermap__sequentually__ne__bot,axiom,
    ! [A: $tType,F3: fun(nat,A)] : filtermap(nat,A,F3,at_top(nat)) != bot_bot(filter(A)) ).

% filtermap_sequentually_ne_bot
tff(fact_5638_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( inj_on(B,A,F3,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_yw(set(A),fun(fun(B,A),fun(B,$o)),A5),F3))) ) ) ).

% finite_inverse_image
tff(fact_5639_finite__UNIV__surj__inj,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),set(A),image2(A,A,F3),top_top(set(A))) = top_top(set(A)) )
       => inj_on(A,A,F3,top_top(set(A))) ) ) ).

% finite_UNIV_surj_inj
tff(fact_5640_finite__UNIV__inj__surj,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( inj_on(A,A,F3,top_top(set(A)))
       => ( aa(set(A),set(A),image2(A,A,F3),top_top(set(A))) = top_top(set(A)) ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_5641_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),A10: set(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ? [F7: fun(A,B)] :
            ( inj_on(A,B,F7,A5)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A5)),A10) )
      <=> ? [G7: fun(B,A)] : aa(set(B),set(A),image2(B,A,G7),A10) = A5 ) ) ).

% inj_on_iff_surj
tff(fact_5642_finite__surj__inj,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),image2(A,A,F3),A5))
       => inj_on(A,A,F3,A5) ) ) ).

% finite_surj_inj
tff(fact_5643_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B5: set(B)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B5)
       => ( aa(set(B),$o,finite_finite2(B),B5)
         => aa(set(A),$o,finite_finite2(A),A5) ) ) ) ).

% inj_on_finite
tff(fact_5644_endo__inj__surj,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image2(A,A,F3),A5)),A5)
       => ( inj_on(A,A,F3,A5)
         => ( aa(set(A),set(A),image2(A,A,F3),A5) = A5 ) ) ) ) ).

% endo_inj_surj
tff(fact_5645_eq__card__imp__inj__on,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F3),A5)) = aa(set(A),nat,finite_card(A),A5) )
       => inj_on(A,B,F3,A5) ) ) ).

% eq_card_imp_inj_on
tff(fact_5646_inj__on__iff__eq__card,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( inj_on(A,B,F3,A5)
      <=> ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F3),A5)) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_5647_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F3),A5))),aa(set(B),nat,finite_card(B),A5))
     => ~ inj_on(B,A,F3,A5) ) ).

% pigeonhole
tff(fact_5648_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: A,Xb: A,B2: A,F3: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
             => ( inj_on(A,B,F3,set_or1337092689740270186AtMost(A,A2,B2))
               => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),aa(A,B,F3,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,B2)) )
                  | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,B2)),aa(A,B,F3,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,A2)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_5649_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I6: set(A),F3: fun(B,C),A5: fun(A,set(B))] :
      ( ( I6 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( member(A,I3,I6)
           => inj_on(B,C,F3,aa(A,set(B),A5,I3)) )
       => inj_on(B,C,F3,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I6))) ) ) ).

% inj_on_INTER
tff(fact_5650_closed__Collect__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F3)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G3)
           => topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_yx(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3))) ) ) ) ).

% closed_Collect_le
tff(fact_5651_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S2: set(A),T2: set(B),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),T2)
       => ( ( aa(set(A),nat,finite_card(A),S2) = aa(set(B),nat,finite_card(B),T2) )
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),S2)),T2)
           => ( ! [X4: B] :
                  ( member(B,X4,T2)
                 => ? [Xa2: A] :
                      ( member(A,Xa2,S2)
                      & ( aa(A,B,F3,Xa2) = X4 ) ) )
            <=> inj_on(A,B,F3,S2) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_5652_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(B),G3: fun(B,A)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B5)
       => ( inj_on(B,A,G3,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G3),B5)),A5)
           => ( aa(set(A),$o,finite_finite2(A),A5)
             => ( aa(set(B),$o,finite_finite2(B),B5)
               => ( aa(set(A),nat,finite_card(A),A5) = aa(set(B),nat,finite_card(B),B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_5653_t4__space,axiom,
    ! [A: $tType] :
      ( topological_t4_space(A)
     => ! [S2: set(A),T2: set(A)] :
          ( topolo7761053866217962861closed(A,S2)
         => ( topolo7761053866217962861closed(A,T2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T2) = bot_bot(set(A)) )
             => ? [U5: set(A),V6: set(A)] :
                  ( topolo1002775350975398744n_open(A,U5)
                  & topolo1002775350975398744n_open(A,V6)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),U5)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),V6)
                  & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V6) = bot_bot(set(A)) ) ) ) ) ) ) ).

% t4_space
tff(fact_5654_t3__space,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [S2: set(A),Yb: A] :
          ( topolo7761053866217962861closed(A,S2)
         => ( ~ member(A,Yb,S2)
           => ? [U5: set(A),V6: set(A)] :
                ( topolo1002775350975398744n_open(A,U5)
                & topolo1002775350975398744n_open(A,V6)
                & member(A,Yb,U5)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),V6)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V6) = bot_bot(set(A)) ) ) ) ) ) ).

% t3_space
tff(fact_5655_continuous__on__closed__Union,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [I6: set(A),U2: fun(A,set(B)),F3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I6)
         => ( ! [I3: A] :
                ( member(A,I3,I6)
               => topolo7761053866217962861closed(B,aa(A,set(B),U2,I3)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I6)
                 => topolo81223032696312382ous_on(B,C,aa(A,set(B),U2,I3),F3) )
             => 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)),F3) ) ) ) ) ).

% continuous_on_closed_Union
tff(fact_5656_Lim__in__closed__set,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F3: fun(B,A),F4: filter(B),L: A] :
          ( topolo7761053866217962861closed(A,S2)
         => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_yy(set(A),fun(fun(B,A),fun(B,$o)),S2),F3),F4)
           => ( ( F4 != bot_bot(filter(B)) )
             => ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4)
               => member(A,L,S2) ) ) ) ) ) ).

% Lim_in_closed_set
tff(fact_5657_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5))
         => ? [F2: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A5)),B5)
              & inj_on(A,B,F2,A5) ) ) ) ) ).

% card_le_inj
tff(fact_5658_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(B)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A5)),B5)
       => ( aa(set(B),$o,finite_finite2(B),B5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ) ) ) ).

% card_inj_on_le
tff(fact_5659_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( ? [F7: fun(A,B)] :
              ( inj_on(A,B,F7,A5)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A5)),B5) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_5660_log__inj,axiom,
    ! [B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => inj_on(real,real,log(B2),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))) ) ).

% log_inj
tff(fact_5661_compact__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U2: set(A)] :
          ( topolo2193935891317330818ompact(A,U2)
        <=> ! [A8: set(set(A))] :
              ( ! [X4: set(A)] :
                  ( member(set(A),X4,A8)
                 => topolo7761053866217962861closed(A,X4) )
             => ( ! [B10: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),B10),A8)
                   => ( aa(set(set(A)),$o,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_5662_compact__imp__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F4: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [T4: set(A)] :
                ( member(set(A),T4,F4)
               => topolo7761053866217962861closed(A,T4) )
           => ( ! [F11: set(set(A))] :
                  ( aa(set(set(A)),$o,finite_finite2(set(A)),F11)
                 => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),F11),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_5663_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( transitive_rtrancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_yg(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_5664_uniformly__continuous__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S: set(A),F3: fun(A,B),E5: fun(product_prod(B,B),$o)] :
          ( topolo6026614971017936543ous_on(A,B,S,F3)
         => ( eventually(product_prod(B,B),E5,topolo7806501430040627800ormity(B))
           => eventually(product_prod(A,A),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(product_prod(B,B),$o),fun(A,fun(A,$o)),aa(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o))),aTP_Lamp_yz(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o)))),S),F3),E5)),topolo7806501430040627800ormity(A)) ) ) ) ).

% uniformly_continuous_onD
tff(fact_5665_measure__function__int,axiom,
    fun_is_measure(int,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))) ).

% measure_function_int
tff(fact_5666_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X5: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_za(fun(A,B),fun(A,product_prod(A,B)),F3),X5) ).

% inj_on_convol_ident
tff(fact_5667_inj__prod__encode,axiom,
    ! [A5: set(product_prod(nat,nat))] : inj_on(product_prod(nat,nat),nat,nat_prod_encode,A5) ).

% inj_prod_encode
tff(fact_5668_inj__int__encode,axiom,
    ! [A5: set(int)] : inj_on(int,nat,nat_int_encode,A5) ).

% inj_int_encode
tff(fact_5669_inj__Suc,axiom,
    ! [N3: set(nat)] : inj_on(nat,nat,suc,N3) ).

% inj_Suc
tff(fact_5670_inj__Some,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,option(A),some(A),A5) ).

% inj_Some
tff(fact_5671_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N3: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N3) ) ).

% inj_on_of_nat
tff(fact_5672_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_5673_measure__fst,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,nat)] :
      ( fun_is_measure(A,F3)
     => fun_is_measure(product_prod(A,B),aTP_Lamp_zb(fun(A,nat),fun(product_prod(A,B),nat),F3)) ) ).

% measure_fst
tff(fact_5674_is__measure_Osimps,axiom,
    ! [A: $tType,A2: fun(A,nat)] :
      ( fun_is_measure(A,A2)
    <=> ? [X_13: fun(A,nat)] : A2 = X_13 ) ).

% is_measure.simps
tff(fact_5675_is__measure__trivial,axiom,
    ! [A: $tType,F3: fun(A,nat)] : fun_is_measure(A,F3) ).

% is_measure_trivial
tff(fact_5676_measure__snd,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,nat)] :
      ( fun_is_measure(A,F3)
     => fun_is_measure(product_prod(B,A),aTP_Lamp_zc(fun(A,nat),fun(product_prod(B,A),nat),F3)) ) ).

% measure_snd
tff(fact_5677_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_rtrancl(A,R2))
     => ( ( A22 != A1 )
       => ~ ! [B4: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4),transitive_rtrancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22),R2) ) ) ) ).

% rtrancl.cases
tff(fact_5678_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_rtrancl(A,R2))
    <=> ( ? [A7: A] :
            ( ( A1 = A7 )
            & ( A22 = A7 ) )
        | ? [A7: A,B7: A,C4: A] :
            ( ( A1 = A7 )
            & ( A22 = C4 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),transitive_rtrancl(A,R2))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C4),R2) ) ) ) ).

% rtrancl.simps
tff(fact_5679_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A2: A,R2: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2),transitive_rtrancl(A,R2)) ).

% rtrancl.rtrancl_refl
tff(fact_5680_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_rtrancl(A,R2)) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_5681_rtranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R2))
     => ( ( A2 != B2 )
       => ~ ! [Y: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y),transitive_rtrancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),B2),R2) ) ) ) ).

% rtranclE
tff(fact_5682_rtrancl__trans,axiom,
    ! [A: $tType,Xb: A,Yb: A,R2: set(product_prod(A,A)),Z: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_rtrancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Yb),Z),transitive_rtrancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),transitive_rtrancl(A,R2)) ) ) ).

% rtrancl_trans
tff(fact_5683_rtrancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P2: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R2))
     => ( aa(A,$o,P2,A2)
       => ( ! [Y: A,Z2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y),transitive_rtrancl(A,R2))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
               => ( aa(A,$o,P2,Y)
                 => aa(A,$o,P2,Z2) ) ) )
         => aa(A,$o,P2,B2) ) ) ) ).

% rtrancl_induct
tff(fact_5684_converse__rtranclE,axiom,
    ! [A: $tType,Xb: A,Z: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),transitive_rtrancl(A,R2))
     => ( ( Xb != Z )
       => ~ ! [Y: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R2)
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z),transitive_rtrancl(A,R2)) ) ) ) ).

% converse_rtranclE
tff(fact_5685_converse__rtrancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P2: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R2))
     => ( aa(A,$o,P2,B2)
       => ( ! [Y: A,Z2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),B2),transitive_rtrancl(A,R2))
               => ( aa(A,$o,P2,Z2)
                 => aa(A,$o,P2,Y) ) ) )
         => aa(A,$o,P2,A2) ) ) ) ).

% converse_rtrancl_induct
tff(fact_5686_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_rtrancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_rtrancl(A,R2)) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_5687_tranclD,axiom,
    ! [A: $tType,Xb: A,Yb: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_trancl(A,R))
     => ? [Z2: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z2),R)
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Yb),transitive_rtrancl(A,R)) ) ) ).

% tranclD
tff(fact_5688_rtranclD,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R))
     => ( ( A2 = B2 )
        | ( ( A2 != B2 )
          & 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_5689_tranclD2,axiom,
    ! [A: $tType,Xb: A,Yb: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_trancl(A,R))
     => ? [Z2: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z2),transitive_rtrancl(A,R))
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Yb),R) ) ) ).

% tranclD2
tff(fact_5690_trancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R2)) ) ).

% trancl_into_rtrancl
tff(fact_5691_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,Xb: A,Yb: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_rtrancl(A,R))
    <=> ( ( Xb = Yb )
        | ( ( Xb != Yb )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_trancl(A,R)) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_5692_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R2)) ) ) ).

% rtrancl_into_trancl1
tff(fact_5693_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_rtrancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R2)) ) ) ).

% rtrancl_into_trancl2
tff(fact_5694_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,Xb: A,Yb: A,R2: set(product_prod(A,A)),Z: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_rtrancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Yb),Z),transitive_trancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z),transitive_trancl(A,R2)) ) ) ).

% rtrancl_trancl_trancl
tff(fact_5695_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_rtrancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R2)) ) ) ).

% trancl_rtrancl_trancl
tff(fact_5696_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R2))
     => ( aa(B,$o,aa(A,fun(B,$o),P2,Bx),By)
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R2)
             => ( 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))
               => ( aa(B,$o,aa(A,fun(B,$o),P2,Aa2),Ba)
                 => aa(B,$o,aa(A,fun(B,$o),P2,A4),B4) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P2,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_5697_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xaa: A,Xb: B,Za: A,Zb: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xaa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb)),transitive_rtrancl(product_prod(A,B),R2))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xaa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
       => ~ ! [A4: A,B4: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xaa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),R2)
             => ~ 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_5698_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R2))
     => ( aa(B,$o,aa(A,fun(B,$o),P2,Ax),Ay)
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),transitive_rtrancl(product_prod(A,B),R2))
             => ( 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)
               => ( aa(B,$o,aa(A,fun(B,$o),P2,A4),B4)
                 => aa(B,$o,aa(A,fun(B,$o),P2,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P2,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_5699_inj__singleton,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,set(A),aTP_Lamp_mu(A,set(A)),A5) ).

% inj_singleton
tff(fact_5700_inj__on__diff__nat,axiom,
    ! [N3: set(nat),K2: nat] :
      ( ! [N2: nat] :
          ( member(nat,N2,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N2) )
     => inj_on(nat,nat,aTP_Lamp_lm(nat,fun(nat,nat),K2),N3) ) ).

% inj_on_diff_nat
tff(fact_5701_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A5: 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_yl(A,fun(B,product_prod(B,A)))),A5) ).

% swap_inj_on
tff(fact_5702_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_5703_measure__size,axiom,
    ! [A: $tType] :
      ( size(A)
     => fun_is_measure(A,size_size(A)) ) ).

% measure_size
tff(fact_5704_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_ze(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_5705_range__inj__infinite,axiom,
    ! [A: $tType,F3: fun(nat,A)] :
      ( inj_on(nat,A,F3,top_top(set(nat)))
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F3),top_top(set(nat)))) ) ).

% range_inj_infinite
tff(fact_5706_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [N2: nat,F2: fun(nat,A)] :
          ( ( A5 = aa(set(nat),set(A),image2(nat,A,F2),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),N2))) )
          & inj_on(nat,A,F2,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_5707_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [F2: fun(A,nat),N2: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F2),A5) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),N2)) )
          & inj_on(A,nat,F2,A5) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_5708_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X3: nat] :
            ( member(nat,X3,I6)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs)) )
       => inj_on(nat,A,nth(A,Xs),I6) ) ) ).

% inj_on_nth
tff(fact_5709_infinite__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ? [F2: fun(nat,A)] :
          ( inj_on(nat,A,F2,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat)))),S2) ) ) ).

% infinite_countable_subset
tff(fact_5710_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
    <=> ? [F7: fun(nat,A)] :
          ( inj_on(nat,A,F7,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F7),top_top(set(nat)))),S2) ) ) ).

% infinite_iff_countable_subset
tff(fact_5711_inj__on__funpow__least,axiom,
    ! [A: $tType,Nb: nat,F3: fun(A,A),S: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),S) = S )
     => ( ! [M: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3),S) != S ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_zf(fun(A,A),fun(A,fun(nat,A)),F3),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% inj_on_funpow_least
tff(fact_5712_uniformly__continuous__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [S: set(A),F3: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S,F3)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X4: A] :
                      ( member(A,X4,S)
                     => ! [Xa2: A] :
                          ( member(A,Xa2,S)
                         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa2,X4)),D6)
                           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,Xa2),aa(A,B,F3,X4))),E4) ) ) ) ) ) ) ) ).

% uniformly_continuous_on_def
tff(fact_5713_isUCont__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F3: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,top_top(set(A)),F3)
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X4: A,Y4: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,Y4)),S7)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X4),aa(A,B,F3,Y4))),R5) ) ) ) ) ) ).

% isUCont_def
tff(fact_5714_surj__int__decode,axiom,
    aa(set(nat),set(int),image2(nat,int,nat_int_decode),top_top(set(nat))) = top_top(set(int)) ).

% surj_int_decode
tff(fact_5715_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F3: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_zg(fun(B,list(A)),fun(B,set(A)),F3)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_5716_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,Nb: int,S2: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_zh(int,fun(A,A),Nb),aa(int,fun(A,A),aTP_Lamp_zi(A,fun(int,fun(A,A)),Xb),Nb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_power_int'
tff(fact_5717_power__int__1__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int] : power_int(A,one_one(A),Nb) = one_one(A) ) ).

% power_int_1_left
tff(fact_5718_power__int__mult__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: num,Nb: num] : power_int(A,power_int(A,Xb,aa(num,int,numeral_numeral(int),Mb)),aa(num,int,numeral_numeral(int),Nb)) = power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb))) ) ).

% power_int_mult_numeral
tff(fact_5719_power__int__1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Yb: A] : power_int(A,Yb,one_one(int)) = Yb ) ).

% power_int_1_right
tff(fact_5720_power__int__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] : sgn_sgn(A,power_int(A,A2,Nb)) = power_int(A,sgn_sgn(A,A2),Nb) ) ).

% power_int_sgn
tff(fact_5721_bind__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,list(A))] : bind(B,A,nil(B),F3) = nil(A) ).

% bind_simps(1)
tff(fact_5722_int__encode__inverse,axiom,
    ! [Xb: int] : aa(nat,int,nat_int_decode,aa(int,nat,nat_int_encode,Xb)) = Xb ).

% int_encode_inverse
tff(fact_5723_int__decode__inverse,axiom,
    ! [Nb: nat] : aa(int,nat,nat_int_encode,aa(nat,int,nat_int_decode,Nb)) = Nb ).

% int_decode_inverse
tff(fact_5724_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W2: num,Yb: A,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Yb),Mb) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W2),Mb)),power_int(A,Yb,Mb)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_5725_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,W2: num,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(num,A,numeral_numeral(A),W2)),Mb) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),power_int(A,aa(num,A,numeral_numeral(A),W2),Mb)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_5726_power__int__0__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Mb: int] :
          ( ( Mb != zero_zero(int) )
         => ( power_int(A,zero_zero(A),Mb) = zero_zero(A) ) ) ) ).

% power_int_0_left
tff(fact_5727_power__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] :
          ( ( power_int(A,Xb,Nb) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Nb != zero_zero(int) ) ) ) ) ).

% power_int_eq_0_iff
tff(fact_5728_power__int__0__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A] : power_int(A,Xb,zero_zero(int)) = one_one(A) ) ).

% power_int_0_right
tff(fact_5729_abs__power__int__minus,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] : aa(A,A,abs_abs(A),power_int(A,aa(A,A,uminus_uminus(A),A2),Nb)) = aa(A,A,abs_abs(A),power_int(A,A2,Nb)) ) ).

% abs_power_int_minus
tff(fact_5730_power__int__of__nat,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: nat] : power_int(A,Xb,aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,A,power_power(A,Xb),Nb) ) ).

% power_int_of_nat
tff(fact_5731_power__int__numeral,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: num] : power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb)) = aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% power_int_numeral
tff(fact_5732_power__int__minus1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Yb: A] : power_int(A,Yb,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Yb) ) ).

% power_int_minus1_right
tff(fact_5733_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Mb))),power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb))) = power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb))) ) ).

% power_int_add_numeral
tff(fact_5734_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: num,Nb: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Mb))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)))),B2) ) ).

% power_int_add_numeral2
tff(fact_5735_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,Nb)),power_int(A,B2,Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_5736_inj__int__decode,axiom,
    ! [A5: set(nat)] : inj_on(nat,int,nat_int_decode,A5) ).

% inj_int_decode
tff(fact_5737_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),power_int(A,Xb,Nb)) ) ) ).

% zero_le_power_int
tff(fact_5738_power__int__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : power_int(A,divide_divide(A,one_one(A),Xb),Nb) = divide_divide(A,one_one(A),power_int(A,Xb,Nb)) ) ).

% power_int_one_over
tff(fact_5739_power__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] : aa(A,A,abs_abs(A),power_int(A,A2,Nb)) = power_int(A,aa(A,A,abs_abs(A),A2),Nb) ) ).

% power_int_abs
tff(fact_5740_power__int__mult,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: int,Nb: int] : power_int(A,Xb,aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb)) = power_int(A,power_int(A,Xb,Mb),Nb) ) ).

% power_int_mult
tff(fact_5741_power__int__commutes,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Nb)),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),power_int(A,Xb,Nb)) ) ).

% power_int_commutes
tff(fact_5742_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Yb: A,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Yb),Mb) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),power_int(A,Yb,Mb)) ) ).

% power_int_mult_distrib
tff(fact_5743_int__decode__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa(nat,int,nat_int_decode,Xb) = aa(nat,int,nat_int_decode,Yb) )
    <=> ( Xb = Yb ) ) ).

% int_decode_eq
tff(fact_5744_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Yb: A,Mb: int] : power_int(A,divide_divide(A,Xb,Yb),Mb) = divide_divide(A,power_int(A,Xb,Mb),power_int(A,Yb,Mb)) ) ).

% power_int_divide_distrib
tff(fact_5745_power__int__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : power_int(A,aa(A,A,inverse_inverse(A),Xb),Nb) = aa(A,A,inverse_inverse(A),power_int(A,Xb,Nb)) ) ).

% power_int_inverse
tff(fact_5746_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),power_int(A,Xb,Nb)) ) ) ).

% zero_less_power_int
tff(fact_5747_power__int__not__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Nb = zero_zero(int) ) )
         => ( power_int(A,Xb,Nb) != zero_zero(A) ) ) ) ).

% power_int_not_zero
tff(fact_5748_power__int__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : power_int(A,Xb,aa(int,int,uminus_uminus(int),Nb)) = aa(A,A,inverse_inverse(A),power_int(A,Xb,Nb)) ) ).

% power_int_minus
tff(fact_5749_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F3: fun(A,B),Nb: int] :
          ( topolo81223032696312382ous_on(A,B,S,F3)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( aa(A,B,F3,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aa(int,fun(A,B),aTP_Lamp_zj(fun(A,B),fun(int,fun(A,B)),F3),Nb)) ) ) ) ).

% continuous_on_power_int
tff(fact_5750_list__bind__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F3: fun(A,list(B)),G3: fun(A,list(B))] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,list(B),F3,X3) = aa(A,list(B),G3,X3) ) )
       => ( bind(A,B,Xs,F3) = bind(A,B,Ys,G3) ) ) ) ).

% list_bind_cong
tff(fact_5751_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Mb: int] :
          power_int(A,zero_zero(A),Mb) = $ite(Mb = zero_zero(int),one_one(A),zero_zero(A)) ) ).

% power_int_0_left_If
tff(fact_5752_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N3: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,Nb)),power_int(A,A2,N3)) ) ) ) ).

% power_int_increasing
tff(fact_5753_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N3: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A2,Nb)),power_int(A,A2,N3)) ) ) ) ).

% power_int_strict_increasing
tff(fact_5754_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Mb != Nb ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),Nb)) = divide_divide(A,power_int(A,Xb,Mb),power_int(A,Xb,Nb)) ) ) ) ).

% power_int_diff
tff(fact_5755_tendsto__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),Nb: int] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aa(int,fun(A,B),aTP_Lamp_zk(fun(A,B),fun(int,fun(A,B)),F3),Nb),topolo7230453075368039082e_nhds(B,power_int(B,A2,Nb)),F4) ) ) ) ).

% tendsto_power_int
tff(fact_5756_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,S: set(A),F3: fun(A,B),Nb: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F3)
         => ( ( aa(A,B,F3,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(int,fun(A,B),aTP_Lamp_zl(fun(A,B),fun(int,fun(A,B)),F3),Nb)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_5757_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),Xb: A,S: set(A),Nb: int] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,B,F3,Xb) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_zm(fun(A,B),fun(int,fun(A,B)),F3),Nb),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% differentiable_power_int
tff(fact_5758_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F3: fun(A,B),Nb: int] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( ( aa(A,B,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aa(int,fun(A,B),aTP_Lamp_zl(fun(A,B),fun(int,fun(A,B)),F3),Nb)) ) ) ) ).

% continuous_power_int
tff(fact_5759_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N3: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A2,N3)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_5760_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Nb)),power_int(A,Yb,Nb)) ) ) ) ) ).

% power_int_mono
tff(fact_5761_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,B2,Nb)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_5762_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),power_int(A,Xb,Nb)) ) ) ) ).

% one_le_power_int
tff(fact_5763_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),power_int(A,A2,Nb)) ) ) ) ).

% one_less_power_int
tff(fact_5764_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),Nb) != zero_zero(int) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),power_int(A,Xb,Nb)) ) ) ) ).

% power_int_add
tff(fact_5765_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,B2,Nb)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_antimono
tff(fact_5766_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A2,Nb)),power_int(A,B2,Nb)) ) ) ) ) ).

% power_int_strict_mono
tff(fact_5767_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Nb)),one_one(A)) ) ) ) ) ).

% power_int_le_one
tff(fact_5768_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N3: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => ( ( ( A2 != zero_zero(A) )
                  | ( N3 != zero_zero(int) )
                  | ( Nb = zero_zero(int) ) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,N3)),power_int(A,A2,Nb)) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_5769_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Mb)),power_int(A,Xb,Nb))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Mb),Nb) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_5770_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,Xb,Mb)),power_int(A,Xb,Nb))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Mb),Nb) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_5771_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Nb != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int)))),Xb) = power_int(A,Xb,Nb) ) ) ) ).

% power_int_minus_mult
tff(fact_5772_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Mb != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),power_int(A,Xb,Mb)) ) ) ) ).

% power_int_add_1'
tff(fact_5773_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Mb != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),Xb) ) ) ) ).

% power_int_add_1
tff(fact_5774_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: int] :
          power_int(A,Xb,Nb) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,A,power_power(A,Xb),aa(int,nat,nat2,Nb)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),Xb)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Nb)))) ) ).

% power_int_def
tff(fact_5775_powr__real__of__int_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( ( Xb != zero_zero(real) )
          | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb) )
       => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),Nb)) = power_int(real,Xb,Nb) ) ) ) ).

% powr_real_of_int'
tff(fact_5776_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,Xb: A,S: set(A),Nb: int] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F3,Xb) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_zn(fun(A,A),fun(int,fun(A,A)),F3),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Nb)),power_int(A,aa(A,A,F3,Xb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int))))),D3),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_power_int
tff(fact_5777_has__derivative__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(B,A),Xb: B,F9: fun(B,A),S2: set(B),Nb: int] :
          ( ( aa(B,A,F3,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F3,F9,topolo174197925503356063within(B,Xb,S2))
           => has_derivative(B,A,aa(int,fun(B,A),aTP_Lamp_zo(fun(B,A),fun(int,fun(B,A)),F3),Nb),aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_zp(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),F3),Xb),F9),Nb),topolo174197925503356063within(B,Xb,S2)) ) ) ) ).

% has_derivative_power_int
tff(fact_5778_power__int__numeral__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: num,Nb: num] : power_int(A,aa(num,A,numeral_numeral(A),Mb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),pow(Mb,Nb))) ) ).

% power_int_numeral_neg_numeral
tff(fact_5779_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),product_apsnd(B,C,A,F3),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F3,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_5780_inj__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),top_top(set(product_prod(A,B))))
    <=> inj_on(A,C,F3,top_top(set(A))) ) ).

% inj_apfst
tff(fact_5781_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F3: fun(C,A),Xb: C,Yb: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Xb),Yb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,Xb)),Yb) ).

% apfst_conv
tff(fact_5782_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),Xb: A,Yb: C] : aa(product_prod(A,C),product_prod(A,B),product_apsnd(C,B,A,F3),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Xb),Yb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),aa(C,B,F3,Yb)) ).

% apsnd_conv
tff(fact_5783_apfst__eq__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),Xb: product_prod(C,B),G3: fun(C,A)] :
      ( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),Xb) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G3),Xb) )
    <=> ( aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),Xb)) = aa(C,A,G3,aa(product_prod(C,B),C,product_fst(C,B),Xb)) ) ) ).

% apfst_eq_conv
tff(fact_5784_fst__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),Xb: product_prod(C,B)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),Xb)) = aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),Xb)) ).

% fst_apfst
tff(fact_5785_snd__apfst,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,B),Xb: product_prod(C,A)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,A),product_prod(B,A),product_apfst(C,B,A,F3),Xb)) = aa(product_prod(C,A),A,product_snd(C,A),Xb) ).

% snd_apfst
tff(fact_5786_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xb: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),product_apsnd(C,B,A,F3),Xb)) = aa(product_prod(A,C),A,product_fst(A,C),Xb) ).

% fst_apsnd
tff(fact_5787_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xb: product_prod(A,C),G3: fun(C,B)] :
      ( ( aa(product_prod(A,C),product_prod(A,B),product_apsnd(C,B,A,F3),Xb) = aa(product_prod(A,C),product_prod(A,B),product_apsnd(C,B,A,G3),Xb) )
    <=> ( aa(C,B,F3,aa(product_prod(A,C),C,product_snd(A,C),Xb)) = aa(C,B,G3,aa(product_prod(A,C),C,product_snd(A,C),Xb)) ) ) ).

% apsnd_eq_conv
tff(fact_5788_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xb: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),product_apsnd(C,A,B,F3),Xb)) = aa(C,A,F3,aa(product_prod(B,C),C,product_snd(B,C),Xb)) ).

% snd_apsnd
tff(fact_5789_snd__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),comp(product_prod(C,B),B,product_prod(A,B),product_snd(C,B)),product_apfst(A,C,B,F3)) = product_snd(A,B) ).

% snd_comp_apfst
tff(fact_5790_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C)),product_apsnd(B,C,A,F3)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_5791_apfst__id,axiom,
    ! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).

% apfst_id
tff(fact_5792_apsnd__id,axiom,
    ! [B: $tType,A: $tType] : product_apsnd(B,B,A,id(B)) = id(product_prod(A,B)) ).

% apsnd_id
tff(fact_5793_fst__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),comp(product_prod(C,B),C,product_prod(A,B),product_fst(C,B)),product_apfst(A,C,B,F3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_apfst
tff(fact_5794_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),comp(product_prod(A,C),C,product_prod(A,B),product_snd(A,C)),product_apsnd(B,C,A,F3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),F3),product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_5795_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),Xb: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(C,D),product_prod(C,B),product_apsnd(D,B,C,G3),Xb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),Xb))),aa(D,B,G3,aa(product_prod(C,D),D,product_snd(C,D),Xb))) ).

% apfst_apsnd
tff(fact_5796_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),Xb: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),product_apsnd(C,B,A,F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),Xb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G3,aa(product_prod(D,C),D,product_fst(D,C),Xb))),aa(C,B,F3,aa(product_prod(D,C),C,product_snd(D,C),Xb))) ).

% apsnd_apfst
tff(fact_5797_apsnd__apfst__commute,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),P: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),product_apsnd(C,B,A,F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),P)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G3),aa(product_prod(D,C),product_prod(D,B),product_apsnd(C,B,D,F3),P)) ).

% apsnd_apfst_commute
tff(fact_5798_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(C,B),G3: fun(D,C),Xb: product_prod(A,D)] : aa(product_prod(A,C),product_prod(A,B),product_apsnd(C,B,A,F3),aa(product_prod(A,D),product_prod(A,C),product_apsnd(D,C,A,G3),Xb)) = aa(product_prod(A,D),product_prod(A,B),product_apsnd(D,B,A,aa(fun(D,C),fun(D,B),comp(C,B,D,F3),G3)),Xb) ).

% apsnd_compose
tff(fact_5799_apfst__compose,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,C),Xb: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G3),Xb)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),comp(C,A,D,F3),G3)),Xb) ).

% apfst_compose
tff(fact_5800_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q2: product_prod(A,B),F3: fun(C,A),P: product_prod(C,B)] :
      ( ( Q2 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),P) )
     => ~ ! [X3: C,Y: B] :
            ( ( P = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X3),Y) )
           => ( Q2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X3)),Y) ) ) ) ).

% apfst_convE
tff(fact_5801_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zq(fun(A,B),fun(B,fun(A,$o)),F3),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_5802_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zr(fun(A,B),fun(B,fun(A,$o)),F3),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_5803_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => member(B,aa(A,B,F3,X3),ring_1_Ints(B)) )
         => member(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5),ring_1_Ints(B)) ) ) ).

% Ints_sum
tff(fact_5804_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A5)
             => member(B,aa(A,B,F3,X3),ring_1_Ints(B)) )
         => member(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5),ring_1_Ints(B)) ) ) ).

% Ints_prod
tff(fact_5805_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( archimedean_frac(A,Xb) = zero_zero(A) )
        <=> member(A,Xb,ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_5806_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Yb: A] :
          ( ( member(A,Xb,ring_1_Ints(A))
            | member(A,Yb,ring_1_Ints(A)) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Yb)) ) ) ) ).

% floor_add2
tff(fact_5807_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),archimedean_frac(A,Xb))
        <=> ~ member(A,Xb,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_5808_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_5809_minus__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A] :
          ( member(A,aa(A,A,uminus_uminus(A),Xb),ring_1_Ints(A))
        <=> member(A,Xb,ring_1_Ints(A)) ) ) ).

% minus_in_Ints_iff
tff(fact_5810_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(A,A,uminus_uminus(A),A2),ring_1_Ints(A)) ) ) ).

% Ints_minus
tff(fact_5811_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_5812_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_5813_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_5814_Ints__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(A,A,abs_abs(A),A2),ring_1_Ints(A)) ) ) ).

% Ints_abs
tff(fact_5815_Ints__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Nb),ring_1_Ints(A)) ) ).

% Ints_of_nat
tff(fact_5816_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: num] : member(A,aa(num,A,numeral_numeral(A),Nb),ring_1_Ints(A)) ) ).

% Ints_numeral
tff(fact_5817_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_5818_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(nat,A,power_power(A,A2),Nb),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_5819_Ints__of__int,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] : member(A,aa(int,A,ring_1_of_int(A),Z),ring_1_Ints(A)) ) ).

% Ints_of_int
tff(fact_5820_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A,P2: fun(A,$o)] :
          ( member(A,Q2,ring_1_Ints(A))
         => ( ! [Z2: int] : aa(A,$o,P2,aa(int,A,ring_1_of_int(A),Z2))
           => aa(A,$o,P2,Q2) ) ) ) ).

% Ints_induct
tff(fact_5821_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A] :
          ( member(A,Q2,ring_1_Ints(A))
         => ~ ! [Z2: int] : Q2 != aa(int,A,ring_1_of_int(A),Z2) ) ) ).

% Ints_cases
tff(fact_5822_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( 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_5823_Nats__subset__Ints,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),semiring_1_Nats(A)),ring_1_Ints(A)) ) ).

% Nats_subset_Ints
tff(fact_5824_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_zs(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_5825_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( 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_5826_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B2),A2)
         => member(A,divide_divide(A,aa(int,A,ring_1_of_int(A),A2),aa(int,A,ring_1_of_int(A),B2)),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_5827_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_zt(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_5828_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_zu(A,$o)) ) ) ).

% Nats_altdef2
tff(fact_5829_Ints__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_Ints(A) = aa(set(int),set(A),image2(int,A,ring_1_of_int(A)),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_5830_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2)),zero_zero(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_5831_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( ( Xb != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),Xb)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5832_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xb)),one_one(A))
           => ( Xb = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5833_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( member(A,Yb,ring_1_Ints(A))
           => ( ( Xb = Yb )
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Yb))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5834_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),Xb)) = $ite(member(A,Xb,ring_1_Ints(A)),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,Xb))) ) ).

% frac_neg
tff(fact_5835_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_5836_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: A] :
          ( ( archimedean_frac(A,Xb) = A2 )
        <=> ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_5837_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5838_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Yb) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Yb)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Yb)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X4,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X9)
                              & ! [X4: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X9) ) ),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zv(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5839_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $let(
                        m2: nat,
                        m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                        ( ! [X4: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_VEBT_valid(X4,n) )
                        & vEBT_VEBT_valid(Summary,m2)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X9)
                            & ! [X4: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X9) ) ),
                            aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zv(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5840_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                    $let(
                      m2: nat,
                      m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                      ( ! [X4: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_VEBT_valid(X4,n) )
                      & vEBT_VEBT_valid(Summary,m2)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X9)
                          & ! [X4: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X9) ) ),
                          aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zv(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5841_ball__empty,axiom,
    ! [A: $tType,P2: fun(A,$o),X: A] :
      ( member(A,X,bot_bot(set(A)))
     => aa(A,$o,P2,X) ) ).

% ball_empty
tff(fact_5842_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zw(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),P2),Q)))
      <=> ! [Y4: A] :
            ( aa(A,$o,P2,Y4)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_zx(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),Y4))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_5843_closed__diagonal,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_zy(product_prod(A,A),$o))) ) ).

% closed_diagonal
tff(fact_5844_finite_Omono,axiom,
    ! [A: $tType] : order_mono(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_zz(fun(set(A),$o),fun(set(A),$o))) ).

% finite.mono
tff(fact_5845_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_al(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_5846_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_aaa(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_5847_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aTP_Lamp_aab(fun(A,$o),fun(fun(A,B),fun(B,$o)),P2),F3))) ) ).

% finite_image_set
tff(fact_5848_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(A,$o),Q: fun(B,$o),F3: fun(A,fun(B,C))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Q))
       => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_aac(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P2),Q),F3))) ) ) ).

% finite_image_set2
tff(fact_5849_option_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),X2: B] : case_option(A,B,F1,F22,aa(B,option(B),some(B),X2)) = aa(B,A,F22,X2) ).

% option.simps(5)
tff(fact_5850_option_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,A)] : case_option(A,B,F1,F22,none(B)) = F1 ).

% option.simps(4)
tff(fact_5851_option_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,B),Option: option(C)] : aa(B,A,H,case_option(B,C,F1,F22,Option)) = case_option(A,C,aa(B,A,H,F1),aa(fun(C,B),fun(C,A),aTP_Lamp_aad(fun(B,A),fun(fun(C,B),fun(C,A)),H),F22),Option) ).

% option.case_distrib
tff(fact_5852_open__diagonal__complement,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aae(product_prod(A,A),$o))) ) ).

% open_diagonal_complement
tff(fact_5853_closed__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aaf(product_prod(A,A),$o))) ) ).

% closed_subdiagonal
tff(fact_5854_closed__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aag(product_prod(A,A),$o))) ) ).

% closed_superdiagonal
tff(fact_5855_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aah(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_5856_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aai(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_5857_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P2: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X3: A] :
            ( member(A,X3,A5)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_zx(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P2),X3),Net) )
       => eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aaj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A5),P2),Net) ) ) ).

% eventually_ball_finite
tff(fact_5858_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P2: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aaj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A5),P2),Net)
      <=> ! [X4: A] :
            ( member(A,X4,A5)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_zx(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P2),X4),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_5859_case__optionE,axiom,
    ! [A: $tType,P2: $o,Q: fun(A,$o),Xb: option(A)] :
      ( case_option($o,A,(P2),Q,Xb)
     => ( ( ( Xb = none(A) )
         => ~ (P2) )
       => ~ ! [Y: A] :
              ( ( Xb = aa(A,option(A),some(A),Y) )
             => ~ aa(A,$o,Q,Y) ) ) ) ).

% case_optionE
tff(fact_5860_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A5) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aak(set(A),fun(A,$o),A5))) ) ).

% Inf_eq_Sup
tff(fact_5861_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A5) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aal(set(A),fun(A,$o),A5))) ) ).

% Sup_eq_Inf
tff(fact_5862_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aam(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_5863_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A)] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S2)
           => ( aa(set(A),A,complete_Inf_Inf(A),S2) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aan(set(A),fun(A,$o),S2))) ) ) ) ) ).

% cInf_cSup
tff(fact_5864_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A)] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S2)
           => ( aa(set(A),A,complete_Sup_Sup(A),S2) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aao(set(A),fun(A,$o),S2))) ) ) ) ) ).

% cSup_cInf
tff(fact_5865_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Deg3: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))),
            $let(
              m2: nat,
              m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Dega),n),
              ( ! [X4: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                 => vEBT_VEBT_valid(X4,n) )
              & vEBT_VEBT_valid(Summarya,m2)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X9)
                  & ! [X4: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X9) ) ),
                  aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zv(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Dega),TreeLista),Summarya),n),m2)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5866_funpow__inj__finite,axiom,
    ! [A: $tType,P: fun(A,A),Xb: A] :
      ( inj_on(A,A,P,top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aap(fun(A,A),fun(A,fun(A,$o)),P),Xb)))
       => ~ ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),P),Xb) != Xb ) ) ) ) ).

% funpow_inj_finite
tff(fact_5867_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X4,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X9)
                              & ! [X4: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X9) ) ),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zv(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5868_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa))
               => ( Xaa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X4,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X9)
                                & ! [X4: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X9) ) ),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zv(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5869_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Yb) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Yb)
                <=> ( Xaa = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Yb)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X4,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X9)
                                & ! [X4: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X9) ) ),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zv(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_5870_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aaq(set(set(A)),fun(set(A),$o),A5))))) ) ).

% finite_Inf_Sup
tff(fact_5871_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aar(set(set(A)),fun(set(A),$o),A5))))) ) ).

% Inf_Sup_le
tff(fact_5872_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aas(set(set(A)),fun(set(A),$o),A5))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A5))) ) ).

% Sup_Inf_le
tff(fact_5873_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q: fun(A,fun(B,C)),F3: fun(D,B)] :
          ( order_mono(A,fun(B,C),Q)
         => order_mono(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aat(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F3)) ) ) ).

% mono_compose
tff(fact_5874_Union__maximal__sets,axiom,
    ! [A: $tType,F13: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),F13)
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aau(set(set(A)),fun(set(A),$o),F13))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F13) ) ) ).

% Union_maximal_sets
tff(fact_5875_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aav(A,$o)) ) ) ).

% Nats_altdef1
tff(fact_5876_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A)] : order_mono(fun(A,$o),fun(A,$o),aTP_Lamp_aaw(fun(A,A),fun(fun(A,$o),fun(A,$o)),F3)) ) ).

% iteratesp.mono
tff(fact_5877_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: num,Nb: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),Mb)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),Mb),Nb)) ) ).

% take_bit_numeral_numeral
tff(fact_5878_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: set(C),F3: fun(C,A),G3: fun(C,B)] : bNF_Greatest_image2(C,A,B,A5,F3,G3) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_aax(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),A5),F3),G3)) ).

% image2_def
tff(fact_5879_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero(nat),Mb) = none(num) ).

% take_bit_num_simps(1)
tff(fact_5880_chain__empty,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).

% chain_empty
tff(fact_5881_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A5: set(A),Z: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z) ) ) ) ).

% ccpo_Sup_least
tff(fact_5882_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A5: set(A),Xb: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
         => ( member(A,Xb,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ).

% ccpo_Sup_upper
tff(fact_5883_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xb: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_5884_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F3: fun(B,A),Xb: B,C3: C,G3: fun(B,C),A5: set(B)] :
      ( ( B2 = aa(B,A,F3,Xb) )
     => ( ( C3 = aa(B,C,G3,Xb) )
       => ( member(B,Xb,A5)
         => member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C3),bNF_Greatest_image2(B,A,C,A5,F3,G3)) ) ) ) ).

% image2_eqI
tff(fact_5885_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: num] :
          ( ( bit_take_bit_num(Mb,Nb) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5886_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A5: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( ( A5 != bot_bot(set(A)) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),A5),A5) ) ) ) ) ).

% in_chain_finite
tff(fact_5887_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = aa(fun(real,$o),set(real),collect(real),aTP_Lamp_aay(real,$o)) ).

% Rats_eq_int_div_nat
tff(fact_5888_UN__le__eq__Un0,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M4),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)))),aa(nat,set(A),M4,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_5889_sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_sup(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xb: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),sup_sup(fun(B,A)),F3),G3),Xb) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F3,Xb)),aa(B,A,G3,Xb)) ) ).

% sup_apply
tff(fact_5890_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ).

% sup.right_idem
tff(fact_5891_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) ) ).

% sup_left_idem
tff(fact_5892_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ).

% sup.left_idem
tff(fact_5893_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Xb) = Xb ) ).

% sup_idem
tff(fact_5894_sup_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),A2) = A2 ) ).

% sup.idem
tff(fact_5895_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z) ) ) ) ).

% le_sup_iff
tff(fact_5896_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% sup.bounded_iff
tff(fact_5897_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),Xb) = top_top(A) ) ).

% sup_top_left
tff(fact_5898_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),top_top(A)) = top_top(A) ) ).

% sup_top_right
tff(fact_5899_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_5900_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_5901_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_5902_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_5903_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = bot_bot(A) )
        <=> ( ( Xb = bot_bot(A) )
            & ( Yb = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_5904_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A,Yb: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) )
        <=> ( ( Xb = bot_bot(A) )
            & ( Yb = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_5905_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),bot_bot(A)) = Xb ) ).

% sup_bot_right
tff(fact_5906_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),Xb) = Xb ) ).

% sup_bot_left
tff(fact_5907_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)) = Xb ) ).

% inf_sup_absorb
tff(fact_5908_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)) = Xb ) ).

% sup_inf_absorb
tff(fact_5909_Un__empty,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> ( ( A5 = bot_bot(set(A)) )
        & ( B5 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_5910_finite__Un,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G5))
    <=> ( aa(set(A),$o,finite_finite2(A),F4)
        & aa(set(A),$o,finite_finite2(A),G5) ) ) ).

% finite_Un
tff(fact_5911_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(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_5912_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: 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_aaz(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C5)) = $ite(C5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C5)))) ).

% UN_simps(3)
tff(fact_5913_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),B5: set(A),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_aba(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C5)) = $ite(C5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),C5))),B5)) ).

% UN_simps(2)
tff(fact_5914_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F3: fun(A,B),A5: A,B5: A] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,A5)),aa(A,B,F3,B5))),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B5))) ) ) ).

% mono_sup
tff(fact_5915_cSUP__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),B5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( ( B5 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),B5))
               => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),B5))) ) ) ) ) ) ) ).

% cSUP_union
tff(fact_5916_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A2: A,B2: A,P2: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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))),P2),Q)))
     => ( ! [X3: A,Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2),transitive_rtrancl(A,P2))
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X3),Q)
             => ( Y = X3 ) ) )
       => 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,P2)) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_5917_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A2: A,B2: A,P2: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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))),P2),Q)))
     => ( ! [X3: A,Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),X3),transitive_rtrancl(A,P2))
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),Q)
             => ( X3 = Y ) ) )
       => 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,P2)) ) ) ).

% rtrancl_Un_separatorE
tff(fact_5918_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5919_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI2
tff(fact_5920_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI1
tff(fact_5921_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_5922_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_5923_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded2
tff(fact_5924_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded1
tff(fact_5925_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_5926_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2) ) ) ) ).

% sup.boundedI
tff(fact_5927_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% sup.boundedE
tff(fact_5928_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = Yb ) ) ) ).

% sup_absorb2
tff(fact_5929_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = Xb ) ) ) ).

% sup_absorb1
tff(fact_5930_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_5931_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_5932_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F3: fun(A,fun(A,A)),Xb: A,Yb: A] :
          ( ! [X3: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,X3),Y))
         => ( ! [X3: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),F3,X3),Y))
           => ( ! [X3: A,Y: A,Z2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,Y),Z2)),X3) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = aa(A,A,aa(A,fun(A,A),F3,Xb),Yb) ) ) ) ) ) ).

% sup_unique
tff(fact_5933_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% sup.orderI
tff(fact_5934_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_5935_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = Yb ) ) ) ).

% le_iff_sup
tff(fact_5936_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Yb: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)),Xb) ) ) ) ).

% sup_least
tff(fact_5937_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),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),C3),D3)) ) ) ) ).

% sup_mono
tff(fact_5938_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_5939_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI2
tff(fact_5940_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI1
tff(fact_5941_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Yb: A,Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)) ) ).

% sup_ge2
tff(fact_5942_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)) ) ).

% sup_ge1
tff(fact_5943_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xb) ) ) ) ).

% le_supI
tff(fact_5944_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xb)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb) ) ) ) ).

% le_supE
tff(fact_5945_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)) ) ).

% inf_sup_ord(3)
tff(fact_5946_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Yb: A,Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)) ) ).

% inf_sup_ord(4)
tff(fact_5947_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z))) ) ).

% distrib_inf_le
tff(fact_5948_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z))) ) ).

% distrib_sup_le
tff(fact_5949_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( ! [X3: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Z2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z)) ) ) ) ).

% distrib_imp1
tff(fact_5950_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( ! [X3: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Z2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z)) ) ) ) ).

% distrib_imp2
tff(fact_5951_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z)) ) ).

% inf_sup_distrib1
tff(fact_5952_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Yb: A,Z: A,Xb: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)),Xb) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Xb)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z),Xb)) ) ).

% inf_sup_distrib2
tff(fact_5953_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z)) ) ).

% sup_inf_distrib1
tff(fact_5954_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Yb: A,Z: A,Xb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Yb),Z)),Xb) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Xb)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z),Xb)) ) ).

% sup_inf_distrib2
tff(fact_5955_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_5956_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_5957_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_5958_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% sup.strict_boundedE
tff(fact_5959_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_5960_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb3
tff(fact_5961_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI2
tff(fact_5962_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI1
tff(fact_5963_Rats__dense__in__real,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
     => ? [X3: real] :
          ( member(real,X3,field_char_0_Rats(real))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),X3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),Yb) ) ) ).

% Rats_dense_in_real
tff(fact_5964_Rats__no__bot__less,axiom,
    ! [Xb: real] :
    ? [X3: real] :
      ( member(real,X3,field_char_0_Rats(real))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),Xb) ) ).

% Rats_no_bot_less
tff(fact_5965_complete__linorder__sup__max,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ( sup_sup(A) = ord_max(A) ) ) ).

% complete_linorder_sup_max
tff(fact_5966_Rats__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => member(A,zero_zero(A),field_char_0_Rats(A)) ) ).

% Rats_0
tff(fact_5967_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F3: fun(A,B),G3: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F3),G3),X) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,X)),aa(A,B,G3,X)) ) ).

% sup_fun_def
tff(fact_5968_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z)) ) ).

% sup_left_commute
tff(fact_5969_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),C3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)) ) ).

% sup.left_commute
tff(fact_5970_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Xb) ) ).

% sup_commute
tff(fact_5971_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: 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),B2),A2) ) ).

% sup.commute
tff(fact_5972_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)) ) ).

% sup_assoc
tff(fact_5973_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)) ) ).

% sup.assoc
tff(fact_5974_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Xb) ) ).

% inf_sup_aci(5)
tff(fact_5975_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)) ) ).

% inf_sup_aci(6)
tff(fact_5976_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z)) ) ).

% inf_sup_aci(7)
tff(fact_5977_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) ) ).

% inf_sup_aci(8)
tff(fact_5978_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_5979_Rats__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,field_char_0_Rats(A))
         => ( member(A,B2,field_char_0_Rats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_add
tff(fact_5980_infinite__Un,axiom,
    ! [A: $tType,S2: set(A),T2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T2))
    <=> ( ~ aa(set(A),$o,finite_finite2(A),S2)
        | ~ aa(set(A),$o,finite_finite2(A),T2) ) ) ).

% infinite_Un
tff(fact_5981_Un__infinite,axiom,
    ! [A: $tType,S2: set(A),T2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T2)) ) ).

% Un_infinite
tff(fact_5982_finite__UnI,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,finite_finite2(A),G5)
       => aa(set(A),$o,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_5983_Rats__infinite,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ~ aa(set(A),$o,finite_finite2(A),field_char_0_Rats(A)) ) ).

% Rats_infinite
tff(fact_5984_Un__empty__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B5) = B5 ).

% Un_empty_left
tff(fact_5985_Un__empty__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),bot_bot(set(A))) = A5 ).

% Un_empty_right
tff(fact_5986_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),bot_bot(A)) = Xb ) ).

% boolean_algebra.disj_zero_right
tff(fact_5987_cSup__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A5)
           => ( ( B5 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,B5)
               => ( 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)),A5),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_5988_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5989_singleton__Un__iff,axiom,
    ! [A: $tType,Xb: A,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) )
    <=> ( ( ( A5 = bot_bot(set(A)) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) )
          & ( B5 = bot_bot(set(A)) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_5990_Un__singleton__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),Xb: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) )
    <=> ( ( ( A5 = bot_bot(set(A)) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) )
          & ( B5 = bot_bot(set(A)) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_5991_insert__is__Un,axiom,
    ! [A: $tType,A2: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) = 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)),insert(A),A2),bot_bot(set(A)))),A5) ).

% insert_is_Un
tff(fact_5992_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or3652927894154168847AtMost(A,Mb,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_5993_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( 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_5994_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Yb) ) ) ).

% sup_shunt
tff(fact_5995_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Yb)),Z)) ) ) ).

% shunt1
tff(fact_5996_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,uminus_uminus(A),Yb))),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Yb),Z)) ) ) ).

% shunt2
tff(fact_5997_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P: A,Q2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q2),R2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P),aa(A,A,uminus_uminus(A),Q2))),R2) ) ) ).

% sup_neg_inf
tff(fact_5998_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A2: A,Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Xb) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Xb) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Yb) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Yb) = top_top(A) )
               => ( Xb = Yb ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_5999_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] :
                  ( member(A,X3,A5)
                 => ( member(A,Y,A5)
                   => member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y),A5) ) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),A5),A5) ) ) ) ) ).

% finite_Sup_in
tff(fact_6000_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))),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)),A5),B5))) ) ).

% less_eq_Inf_inter
tff(fact_6001_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_6002_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_6003_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F3),bot_bot(set(B))))) = A5 ).

% Union_image_empty
tff(fact_6004_card__Un__le,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))) ).

% card_Un_le
tff(fact_6005_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( set_or7035219750837199246ssThan(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I2,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_6006_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or3652927894154168847AtMost(A,Mb,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_6007_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_6008_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_6009_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_6010_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),C5))) = $ite(C5 = bot_bot(set(B)),A5,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_aaz(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C5))) ).

% UN_extend_simps(3)
tff(fact_6011_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),C5: set(B),B5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),C5))),B5) = $ite(C5 = bot_bot(set(B)),B5,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_aba(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C5))) ).

% UN_extend_simps(2)
tff(fact_6012_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Yb) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Yb) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),Xb) = Yb ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_6013_cSup__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( condit941137186595557371_above(A,X5)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),X5)) = $ite(X5 = bot_bot(set(A)),A2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Sup_Sup(A),X5))) ) ) ) ).

% cSup_insert_If
tff(fact_6014_cSup__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X5)
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),X5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ).

% cSup_insert
tff(fact_6015_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_6016_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),B5)) ) ) ) ) ).

% sum.union_inter
tff(fact_6017_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ).

% prod.union_inter
tff(fact_6018_card__Un__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) = 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)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ).

% card_Un_Int
tff(fact_6019_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_6020_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_6021_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_6022_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_6023_cInf__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A5)
           => ( ( B5 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,B5)
               => ( 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)),A5),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6024_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or5935395276787703475ssThan(A,Mb,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_6025_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != 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)),A5),B5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B5)) ) ) ) ) ) ) ).

% Max.union
tff(fact_6026_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_6027_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_6028_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)),insert(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = aa(A,set(A),set_ord_atLeast(A),L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_6029_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_6030_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: A,B5: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_mq(A,fun(nat,A),B5)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B5) ) ).

% SUP_nat_binary
tff(fact_6031_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_6032_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G3),A5))
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G3),A5))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abb(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)),A5)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_6033_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),G3: fun(A,B),B5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( inj_on(A,B,G3,B5)
       => ( ( 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,F3),A5)),aa(set(A),set(B),image2(A,B,G3),B5)) = 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_abc(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F3),A5),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_6034_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_abd(A,fun(A,$o)),aTP_Lamp_abe(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_6035_rtrancl__insert,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abf(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A2),B2),R2)))) ).

% rtrancl_insert
tff(fact_6036_trancl__insert2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abg(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A2),B2),R2)))) ).

% trancl_insert2
tff(fact_6037_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( condit941137186595557371_above(A,A5)
         => ( condit941137186595557371_above(A,B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_6038_cSUP__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),A2: A] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,A2)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ) ).

% cSUP_insert
tff(fact_6039_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,G3,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),B5)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_6040_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% sum_Un
tff(fact_6041_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),B5)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6042_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,G3,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6043_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6044_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_6045_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% sum.union_diff2
tff(fact_6046_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ).

% sum_Un2
tff(fact_6047_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% prod.union_diff2
tff(fact_6048_card__Un__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = 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)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6049_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
    <=> ( inj_on(A,B,F3,A5)
        & inj_on(A,B,F3,B5)
        & ( 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,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_6050_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_6051_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or5935395276787703475ssThan(A,Mb,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_6052_cINF__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),B5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),A5))
           => ( ( B5 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F3),B5))
               => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),B5))) ) ) ) ) ) ) ).

% cINF_union
tff(fact_6053_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_6054_sum__Un__nat,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),B5))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ).

% sum_Un_nat
tff(fact_6055_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_6056_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_6057_trancl__insert,axiom,
    ! [A: $tType,Yb: A,Xb: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Yb),Xb)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abf(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Yb),Xb),R2)))) ).

% trancl_insert
tff(fact_6058_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,F3,X3) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = divide_divide(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ) ).

% prod_Un
tff(fact_6059_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G3: fun(A,B),C5: set(A),B5: set(A),Xb: A] :
      ( inj_on(A,B,G3,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))
       => member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_abh(fun(A,B),fun(set(A),fun(A,fun(B,A))),G3),C5),Xb),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)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_6060_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))))),insert(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_6061_flat__lub__def,axiom,
    ! [A: $tType,B2: A,A5: set(A)] :
      partial_flat_lub(A,B2,A5) = $ite(aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),B2,the(A,aa(set(A),fun(A,$o),aTP_Lamp_abi(A,fun(set(A),fun(A,$o)),B2),A5))) ).

% flat_lub_def
tff(fact_6062_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B)),X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),S2)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),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_6063_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_6064_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_6065_min__strict__def,axiom,
    fun_min_strict = min_ext(product_prod(nat,nat),fun_pair_less) ).

% min_strict_def
tff(fact_6066_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))))),insert(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_6067_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A14: set(B),B1: set(A),F22: fun(C,D),B22: set(C),A25: set(D)] :
      ( ( aa(set(B),set(A),image2(B,A,F1),A14) = B1 )
     => ( inj_on(C,D,F22,B22)
       => ( aa(set(D),$o,aa(set(D),fun(set(D),$o),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,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,A14)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_6068_max__strict__def,axiom,
    fun_max_strict = max_ext(product_prod(nat,nat),fun_pair_less) ).

% max_strict_def
tff(fact_6069_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R))
     => ~ ( aa(set(A),$o,finite_finite2(A),A1)
         => ( aa(set(A),$o,finite_finite2(A),A22)
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X: A] :
                    ( member(A,X,A1)
                   => ? [Xa3: A] :
                        ( member(A,Xa3,A22)
                        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa3),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_6070_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R))
    <=> ( aa(set(A),$o,finite_finite2(A),A1)
        & aa(set(A),$o,finite_finite2(A),A22)
        & ( A22 != bot_bot(set(A)) )
        & ! [X4: A] :
            ( member(A,X4,A1)
           => ? [Xa2: A] :
                ( member(A,Xa2,A22)
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R) ) ) ) ) ).

% max_ext.simps
tff(fact_6071_max__ext_Omax__extI,axiom,
    ! [A: $tType,X5: set(A),Y5: set(A),R: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( aa(set(A),$o,finite_finite2(A),Y5)
       => ( ( Y5 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => ? [Xa: A] :
                    ( member(A,Xa,Y5)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa),R) ) )
           => member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X5),Y5),max_ext(A,R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_6072_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B5: set(A),A5: set(B)] :
      ( ( B5 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,A5,B5) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_6073_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,A5,B5) = bot_bot(set(fun(A,B))) )
    <=> ( ( A5 != bot_bot(set(A)) )
        & ( B5 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_6074_max__ext__def,axiom,
    ! [A: $tType,X: set(product_prod(A,A))] : max_ext(A,X) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),max_extp(A,aTP_Lamp_yb(set(product_prod(A,A)),fun(A,fun(A,$o)),X)))) ).

% max_ext_def
tff(fact_6075_Pow__fold,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( pow2(A,A5) = finite_fold(A,set(set(A)),aTP_Lamp_abj(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))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A5) ) ) ).

% Pow_fold
tff(fact_6076_fold__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),Z: A] : finite_fold(B,A,F3,Z,bot_bot(set(B))) = Z ).

% fold_empty
tff(fact_6077_fold__infinite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,fun(B,B)),Z: B] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( finite_fold(A,B,F3,Z,A5) = Z ) ) ).

% fold_infinite
tff(fact_6078_fold__closed__eq,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B),F3: fun(A,fun(B,B)),G3: fun(A,fun(B,B)),Z: B] :
      ( ! [A4: A,B4: B] :
          ( member(A,A4,A5)
         => ( member(B,B4,B5)
           => ( aa(B,B,aa(A,fun(B,B),F3,A4),B4) = aa(B,B,aa(A,fun(B,B),G3,A4),B4) ) ) )
     => ( ! [A4: A,B4: B] :
            ( member(A,A4,A5)
           => ( member(B,B4,B5)
             => member(B,aa(B,B,aa(A,fun(B,B),G3,A4),B4),B5) ) )
       => ( member(B,Z,B5)
         => ( finite_fold(A,B,F3,Z,A5) = finite_fold(A,B,G3,Z,A5) ) ) ) ) ).

% fold_closed_eq
tff(fact_6079_union__fold__insert,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = finite_fold(A,set(A),insert(A),B5,A5) ) ) ).

% union_fold_insert
tff(fact_6080_sup__Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B5) = finite_fold(A,A,sup_sup(A),B5,A5) ) ) ) ).

% sup_Sup_fold_sup
tff(fact_6081_inf__Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A5)),B5) = finite_fold(A,A,inf_inf(A),B5,A5) ) ) ) ).

% inf_Inf_fold_inf
tff(fact_6082_fold__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: fun(A,B),A5: set(A),F3: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G3,A5)
     => ( finite_fold(B,C,F3,Z,aa(set(A),set(B),image2(A,B,G3),A5)) = finite_fold(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F3),G3),Z,A5) ) ) ).

% fold_image
tff(fact_6083_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,complete_Sup_Sup(A),A5) = finite_fold(A,A,sup_sup(A),bot_bot(A),A5) ) ) ) ).

% Sup_fold_sup
tff(fact_6084_Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,complete_Inf_Inf(A),A5) = finite_fold(A,A,inf_inf(A),top_top(A),A5) ) ) ) ).

% Inf_fold_inf
tff(fact_6085_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G3),zero_zero(A),A5) ) ).

% sum.eq_fold
tff(fact_6086_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = finite_fold(A,A,ord_max(A),Xb,A5) ) ) ) ).

% Max.eq_fold
tff(fact_6087_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),set(B),image2(A,B,F3),A5) = finite_fold(A,set(B),aTP_Lamp_abk(fun(A,B),fun(A,fun(set(B),set(B))),F3),bot_bot(set(B)),A5) ) ) ).

% image_fold_insert
tff(fact_6088_sup__SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),B5: B,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),B5),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F3),B5,A5) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_6089_inf__INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),B5: B,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),B5),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F3),B5,A5) ) ) ) ).

% inf_INF_fold_inf
tff(fact_6090_SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F3),bot_bot(B),A5) ) ) ) ).

% SUP_fold_sup
tff(fact_6091_INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F3),top_top(B),A5) ) ) ) ).

% INF_fold_inf
tff(fact_6092_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: set(A),Xa: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,aTP_Lamp_yb(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X),Xa)
    <=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X),Xa),max_ext(A,R)) ) ).

% max_extp_max_ext_eq
tff(fact_6093_Set__filter__fold,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( filter3(A,P2,A5) = finite_fold(A,set(A),aTP_Lamp_abl(fun(A,$o),fun(A,fun(set(A),set(A))),P2),bot_bot(set(A)),A5) ) ) ).

% Set_filter_fold
tff(fact_6094_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B5: set(A),Xb: B,A5: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( 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_abm(B,fun(A,set(product_prod(B,A))),Xb)),B5))),A5) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_abn(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xb),A5,B5) ) ) ).

% fold_union_pair
tff(fact_6095_finite__filter,axiom,
    ! [A: $tType,S2: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => aa(set(A),$o,finite_finite2(A),filter3(A,P2,S2)) ) ).

% finite_filter
tff(fact_6096_Ball__fold,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X4: A] :
            ( member(A,X4,A5)
           => aa(A,$o,P2,X4) )
      <=> finite_fold(A,$o,aTP_Lamp_abo(fun(A,$o),fun(A,fun($o,$o)),P2),$true,A5) ) ) ).

% Ball_fold
tff(fact_6097_card_Oeq__fold,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = finite_fold(A,nat,aTP_Lamp_abp(A,fun(nat,nat)),zero_zero(nat),A5) ).

% card.eq_fold
tff(fact_6098_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),nil(A),A5) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_6099_inter__Set__filter,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = filter3(A,aTP_Lamp_a(set(A),fun(A,$o),A5),B5) ) ) ).

% inter_Set_filter
tff(fact_6100_Id__on__fold,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( id_on(A,A5) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_abq(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A5) ) ) ).

% Id_on_fold
tff(fact_6101_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(num,nat,numeral_numeral(nat),Mb)) = zero_zero(nat),none(num),aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(num,nat,numeral_numeral(nat),Mb))))) ).

% take_bit_num_def
tff(fact_6102_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),Xb: product_prod(C,A),X5: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,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))),insert(product_prod(C,A)),Xb),bot_bot(set(product_prod(C,A)))),S2)),X5) = 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_abr(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xb)),X5,S2) ) ) ).

% insert_relcomp_union_fold
tff(fact_6103_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_6104_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_6105_finite__relcomp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),S2)
       => aa(set(product_prod(A,C)),$o,finite_finite2(product_prod(A,C)),relcomp(A,B,C,R,S2)) ) ) ).

% finite_relcomp
tff(fact_6106_relpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,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))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),R) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Mb),R),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R)) ).

% relpow_add
tff(fact_6107_relcomp_Ocases,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),relcomp(A,C,B,R2,S))
     => ~ ! [B4: C] :
            ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),B4),R2)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),A22),S) ) ) ).

% relcomp.cases
tff(fact_6108_relcomp_Osimps,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22),relcomp(A,C,B,R2,S))
    <=> ? [A7: A,B7: C,C4: B] :
          ( ( A1 = A7 )
          & ( A22 = C4 )
          & member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A7),B7),R2)
          & member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B7),C4),S) ) ) ).

% relcomp.simps
tff(fact_6109_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R2: set(product_prod(A,B)),C3: C,S: set(product_prod(B,C))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),R2)
     => ( member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3),S)
       => member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),C3),relcomp(A,B,C,R2,S)) ) ) ).

% relcomp.relcompI
tff(fact_6110_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))] :
      ( member(product_prod(A,B),Xz,relcomp(A,C,B,R2,S))
     => ~ ! [X3: A,Y: C,Z2: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z2) )
           => ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X3),Y),R2)
             => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y),Z2),S) ) ) ) ).

% relcompE
tff(fact_6111_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A2: A,C3: B,R2: set(product_prod(A,C)),S: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),C3),relcomp(A,C,B,R2,S))
     => ~ ! [B4: C] :
            ( 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)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),C3),S) ) ) ).

% relcompEpair
tff(fact_6112_union__comp__emptyR,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),B5: set(product_prod(A,A)),C5: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A5,B5) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,A5,C5) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,A5,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))),B5),C5)) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyR
tff(fact_6113_union__comp__emptyL,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),C5: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A5,C5) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,B5,C5) = 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))),A5),B5),C5) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyL
tff(fact_6114_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_6115_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S: set(product_prod(C,B))] : relcomp(A,C,B,R2,S) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_abs(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R2),S))) ).

% relcomp_unfold
tff(fact_6116_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_6117_max__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S2)),R)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,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)))),insert(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_6118_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),one_one(nat))
     => ( num_of_nat(Nb) = one2 ) ) ).

% num_of_nat_One
tff(fact_6119_min__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S2)),R)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,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)))),insert(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_6120_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),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_abu(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_6121_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(num,A,numeral_numeral(A),num_of_nat(Nb)) = $ite(Nb = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% numeral_num_of_nat_unfold
tff(fact_6122_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Nb)) = bit0(num_of_nat(Nb)) ) ) ).

% num_of_nat_double
tff(fact_6123_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),Xb: product_prod(C,A),R: set(product_prod(C,A))] :
      ( aa(set(product_prod(A,B)),$o,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))),insert(product_prod(C,A)),Xb),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_abr(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xb)),relcomp(C,A,B,R,S2),S2) ) ) ).

% insert_relcomp_fold
tff(fact_6124_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(Mb)),num_of_nat(Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_6125_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,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_abw(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_6126_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_abx(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B5)) ) ).

% comp_fun_commute_product_fold
tff(fact_6127_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_aby(fun(B,B),fun(A,fun(B,B)),F3)) ).

% comp_fun_commute_const
tff(fact_6128_comp__fun__commute__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F3)
    <=> ! [Y4: A,X4: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y4)),aa(A,fun(B,B),F3,X4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X4)),aa(A,fun(B,B),F3,Y4)) ) ).

% comp_fun_commute_def
tff(fact_6129_comp__fun__commute_Ocomp__fun__commute,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),Yb: A,Xb: A] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Yb)),aa(A,fun(B,B),F3,Xb)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Xb)),aa(A,fun(B,B),F3,Yb)) ) ) ).

% comp_fun_commute.comp_fun_commute
tff(fact_6130_comp__fun__commute_Ointro,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B))] :
      ( ! [Y: A,X3: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y)),aa(A,fun(B,B),F3,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y))
     => finite6289374366891150609ommute(A,B,F3) ) ).

% comp_fun_commute.intro
tff(fact_6131_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,fun(B,B)),G3: fun(C,A)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => finite6289374366891150609ommute(C,B,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G3)) ) ).

% comp_fun_commute.comp_comp_fun_commute
tff(fact_6132_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P2: fun(A,$o)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_abl(fun(A,$o),fun(A,fun(set(A),set(A))),P2)) ).

% comp_fun_commute_filter_fold
tff(fact_6133_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),G3: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_abz(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F3),G3)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_6134_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),I2: B,B5: 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),A5,I2,B5)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),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)),insert(B),I2),bot_bot(set(B))))))),
        $ite(member(B,I2,J4),B5,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_6135_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),B5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),S2)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = finite_fold(A,B,F3,finite_fold(A,B,F3,Z,A5),B5) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_6136_comp__fun__commute__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,S2)
         => ( member(A,Y,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y)),aa(A,fun(B,B),F3,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y)) ) ) )
     => finite4664212375090638736ute_on(A,B,S2,F3) ) ).

% comp_fun_commute_on.intro
tff(fact_6137_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,Yb: A,G3: fun(C,B)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( member(A,Xb,S2)
       => ( member(A,Yb,S2)
         => ( aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F3,Yb)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F3,Xb)),G3)) = aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F3,Xb)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F3,Yb)),G3)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_6138_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,Yb: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( member(A,Xb,S2)
       => ( member(A,Yb,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Yb)),aa(A,fun(B,B),F3,Xb)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Xb)),aa(A,fun(B,B),F3,Yb)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_6139_comp__fun__commute__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
    <=> ! [X4: A,Y4: A] :
          ( member(A,X4,S2)
         => ( member(A,Y4,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y4)),aa(A,fun(B,B),F3,X4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X4)),aa(A,fun(B,B),F3,Y4)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_6140_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),G3: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => finite4664212375090638736ute_on(A,B,S2,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_abz(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F3),G3)) ) ).

% comp_fun_commute_on.comp_fun_commute_on_funpow
tff(fact_6141_comp__fun__commute__on_Ofun__left__comm,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,Yb: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( member(A,Xb,S2)
       => ( member(A,Yb,S2)
         => ( aa(B,B,aa(A,fun(B,B),F3,Yb),aa(B,B,aa(A,fun(B,B),F3,Xb),Z)) = aa(B,B,aa(A,fun(B,B),F3,Xb),aa(B,B,aa(A,fun(B,B),F3,Yb),Z)) ) ) ) ) ).

% comp_fun_commute_on.fun_left_comm
tff(fact_6142_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C3: B,P2: fun(fun(A,B),$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_aca(fun(A,B),fun(B,fun(A,$o)),F3),C3)))
     => ( aa(fun(A,B),$o,P2,aTP_Lamp_lx(B,fun(A,B),C3))
       => ( ! [A4: A,B4: B,F2: fun(A,B)] :
              ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_acb(B,fun(fun(A,B),fun(A,$o)),C3),F2)))
             => ( ( aa(A,B,F2,A4) = C3 )
               => ( ( B4 != C3 )
                 => ( aa(fun(A,B),$o,P2,F2)
                   => aa(fun(A,B),$o,P2,fun_upd(A,B,F2,A4,B4)) ) ) ) )
         => aa(fun(A,B),$o,P2,F3) ) ) ) ).

% finite_update_induct
tff(fact_6143_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_mr(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_6144_comp__fun__commute__def_H,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F3)
    <=> finite4664212375090638736ute_on(A,B,top_top(set(A)),F3) ) ).

% comp_fun_commute_def'
tff(fact_6145_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),G3: fun(A,fun(B,B)),A5: set(A),S: B,Ta: B,B5: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( finite4664212375090638736ute_on(A,B,S2,G3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( ! [X3: A] :
                  ( member(A,X3,A5)
                 => ( aa(A,fun(B,B),F3,X3) = aa(A,fun(B,B),G3,X3) ) )
             => ( ( S = Ta )
               => ( ( A5 = B5 )
                 => ( finite_fold(A,B,F3,S,A5) = finite_fold(A,B,G3,Ta,B5) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_6146_fold__atLeastAtMost__nat,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      ( finite6289374366891150609ommute(nat,A,F3)
     => ( set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = finite_fold(nat,A,F3,Acc2,set_or1337092689740270186AtMost(nat,A2,B2)) ) ) ).

% fold_atLeastAtMost_nat
tff(fact_6147_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xb: B,Yb: A,A5: set(B)] :
      aa(set(B),set(A),image2(B,A,fun_upd(B,A,F3,Xb,Yb)),A5) = $ite(member(B,Xb,A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),aa(set(B),set(A),image2(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),bot_bot(set(B)))))),aa(set(B),set(A),image2(B,A,F3),A5)) ).

% fun_upd_image
tff(fact_6148_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F3: fun(A,fun(B,B)),G3: fun(C,A),R: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G3),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,F3),G3)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_6149_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,B,aa(A,fun(B,B),F3,Xb),finite_fold(A,B,F3,Z,A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,Xb),Z),A5) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_6150_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,Xb),Z),A5) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_6151_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xb),finite_fold(A,B,F3,Z,A5)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_6152_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_abj(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_6153_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),Xb: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( finite_fold(A,B,F3,Z,A5) = aa(B,B,aa(A,fun(B,B),F3,Xb),finite_fold(A,B,F3,Z,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)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_6154_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xb),finite_fold(A,B,F3,Z,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)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_6155_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I6)) = aa(fun(A,$o),set(A),collect(A),aa(set(nat),fun(A,$o),aTP_Lamp_acc(list(A),fun(set(nat),fun(A,$o)),Xs),I6)) ).

% set_nths
tff(fact_6156_minus__fold__remove,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5) = finite_fold(A,set(A),remove(A),B5,A5) ) ) ).

% minus_fold_remove
tff(fact_6157_nths__nil,axiom,
    ! [A: $tType,A5: set(nat)] : nths(A,nil(A),A5) = nil(A) ).

% nths_nil
tff(fact_6158_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_6159_in__set__nthsD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I6: set(nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),nths(A,Xs,I6)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_nthsD
tff(fact_6160_notin__set__nthsI,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I6: set(nat)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ~ member(A,Xb,aa(list(A),set(A),set2(A),nths(A,Xs,I6))) ) ).

% notin_set_nthsI
tff(fact_6161_distinct__nthsI,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( distinct(A,Xs)
     => distinct(A,nths(A,Xs,I6)) ) ).

% distinct_nthsI
tff(fact_6162_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I6))),aa(list(A),set(A),set2(A),Xs)) ).

% set_nths_subset
tff(fact_6163_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => member(nat,I3,I6) )
     => ( nths(A,Xs,I6) = Xs ) ) ).

% nths_all
tff(fact_6164_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I6)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_acd(list(A),fun(set(nat),fun(nat,$o)),Xs),I6))) ).

% length_nths
tff(fact_6165_remove__code_I1_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xb),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,Xb),Xs)) ).

% remove_code(1)
tff(fact_6166_remove__def,axiom,
    ! [A: $tType,Xb: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xb),A5) = 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)),insert(A),Xb),bot_bot(set(A)))) ).

% remove_def
tff(fact_6167_image__map__upd,axiom,
    ! [B: $tType,A: $tType,Xb: A,A5: set(A),Mb: fun(A,option(B)),Yb: B] :
      ( ~ member(A,Xb,A5)
     => ( aa(set(A),set(option(B)),image2(A,option(B),fun_upd(A,option(B),Mb,Xb,aa(B,option(B),some(B),Yb))),A5) = aa(set(A),set(option(B)),image2(A,option(B),Mb),A5) ) ) ).

% image_map_upd
tff(fact_6168_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),A2: B,B2: A] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F3),top_top(set(B))))
     => aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),fun_upd(B,option(A),F3,A2,aa(A,option(A),some(A),B2))),top_top(set(B)))) ) ).

% finite_range_updI
tff(fact_6169_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),A2: B,B2: A,Xb: B,Yb: A] :
      ( ( aa(B,option(A),fun_upd(B,option(A),Mb,A2,aa(A,option(A),some(A),B2)),Xb) = aa(A,option(A),some(A),Yb) )
    <=> ( ( ( Xb = A2 )
          & ( B2 = Yb ) )
        | ( ( Xb != A2 )
          & ( aa(B,option(A),Mb,Xb) = aa(A,option(A),some(A),Yb) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_6170_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,Ta: fun(B,option(A)),K2: B,Xb: A] :
      ( ( aa(B,option(A),Ta,K2) = aa(A,option(A),some(A),Xb) )
     => ( fun_upd(B,option(A),Ta,K2,aa(A,option(A),some(A),Xb)) = Ta ) ) ).

% map_upd_triv
tff(fact_6171_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),A2: A,Xb: B,Nb: fun(A,option(B)),Yb: B] :
      ( ( fun_upd(A,option(B),Mb,A2,aa(B,option(B),some(B),Xb)) = fun_upd(A,option(B),Nb,A2,aa(B,option(B),some(B),Yb)) )
     => ( Xb = Yb ) ) ).

% map_upd_eqD1
tff(fact_6172_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,Ta: fun(A,option(B)),K2: A,Xb: B] :
      ~ ! [X3: A] : aa(A,option(B),fun_upd(A,option(B),Ta,K2,aa(B,option(B),some(B),Xb)),X3) = none(B) ).

% map_upd_nonempty
tff(fact_6173_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),K2: A,V: B] : graph(A,B,fun_upd(A,option(B),Mb,K2,aa(B,option(B),some(B),V))) = 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))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V)),graph(A,B,fun_upd(A,option(B),Mb,K2,none(B)))) ).

% graph_map_upd
tff(fact_6174_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Xb: A,Yb: B] : restrict_map(A,B,fun_upd(A,option(B),Mb,Xb,aa(B,option(B),some(B),Yb)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))) = restrict_map(A,B,Mb,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_6175_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),X: A] : aa(A,option(B),restrict_map(A,B,Mb,bot_bot(set(A))),X) = none(B) ).

% restrict_map_to_empty
tff(fact_6176_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_ace(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_6177_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Xb: A,Yb: option(B),D5: set(A)] :
      restrict_map(A,B,fun_upd(A,option(B),Mb,Xb,Yb),D5) = $ite(member(A,Xb,D5),fun_upd(A,option(B),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),Xb,Yb),restrict_map(A,B,Mb,D5)) ).

% restrict_fun_upd
tff(fact_6178_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,Xb: A,D5: set(A),Mb: fun(A,option(B)),Yb: option(B)] :
      ( member(A,Xb,D5)
     => ( fun_upd(A,option(B),restrict_map(A,B,Mb,D5),Xb,Yb) = fun_upd(A,option(B),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),Xb,Yb) ) ) ).

% fun_upd_restrict_conv
tff(fact_6179_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),D5: set(A),Xb: A] :
      fun_upd(A,option(B),restrict_map(A,B,Mb,D5),Xb,none(B)) = $ite(member(A,Xb,D5),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),restrict_map(A,B,Mb,D5)) ).

% fun_upd_None_restrict
tff(fact_6180_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K2: A,V: B,Mb: fun(A,option(B)),A5: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V),graph(A,B,restrict_map(A,B,Mb,A5)))
     => member(A,K2,A5) ) ).

% graph_restrictD(1)
tff(fact_6181_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K2: A,V: B,Mb: fun(A,option(B)),A5: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V),graph(A,B,restrict_map(A,B,Mb,A5)))
     => ( aa(A,option(B),Mb,K2) = aa(B,option(B),some(B),V) ) ) ).

% graph_restrictD(2)
tff(fact_6182_in__graphD,axiom,
    ! [A: $tType,B: $tType,K2: A,V: B,Mb: fun(A,option(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V),graph(A,B,Mb))
     => ( aa(A,option(B),Mb,K2) = aa(B,option(B),some(B),V) ) ) ).

% in_graphD
tff(fact_6183_in__graphI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),K2: B,V: A] :
      ( ( aa(B,option(A),Mb,K2) = aa(A,option(A),some(A),V) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),V),graph(B,A,Mb)) ) ).

% in_graphI
tff(fact_6184_graph__def,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B))] : graph(A,B,Mb) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_acf(fun(A,option(B)),fun(product_prod(A,B),$o),Mb)) ).

% graph_def
tff(fact_6185_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),D5: set(A),Xb: A,Yb: option(B)] : fun_upd(A,option(B),restrict_map(A,B,Mb,D5),Xb,Yb) = fun_upd(A,option(B),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),Xb,Yb) ).

% fun_upd_restrict
tff(fact_6186_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),Xb: A] : restrict_map(A,B,F3,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))) = fun_upd(A,option(B),F3,Xb,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_6187_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),D5: set(A),Mb: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D5)
       => ( restrict_map(A,B,map_upds(A,B,Mb,Xs,Ys),D5) = map_upds(A,B,restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(list(A),set(A),set2(A),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_6188_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),Mb,A2) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),Mb,A2,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),ran(B,A,Mb)) ) ) ).

% ran_map_upd
tff(fact_6189_map__upds__apply__nontin,axiom,
    ! [B: $tType,A: $tType,Xb: A,Xs: list(A),F3: fun(A,option(B)),Ys: list(B)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,option(B),map_upds(A,B,F3,Xs,Ys),Xb) = aa(A,option(B),F3,Xb) ) ) ).

% map_upds_apply_nontin
tff(fact_6190_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_acg(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_6191_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,Mb: fun(A,option(B)),Ys: list(B),Yb: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2)
     => ( map_upds(A,B,Mb,Xs,list_update(B,Ys,I2,Yb)) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6192_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A2: A,As: list(A),Mb: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ member(A,A2,aa(list(A),set(A),set2(A),As))
     => ( map_upds(A,B,fun_upd(A,option(B),Mb,A2,aa(B,option(B),some(B),B2)),As,Bs) = fun_upd(A,option(B),map_upds(A,B,Mb,As,Bs),A2,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_6193_ranI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),Mb,A2) = aa(A,option(A),some(A),B2) )
     => member(A,B2,ran(B,A,Mb)) ) ).

% ranI
tff(fact_6194_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Yb: A,Mb: fun(B,option(A)),A5: set(B)] :
      ( member(A,Yb,ran(B,A,restrict_map(B,A,Mb,A5)))
     => ? [X3: B] :
          ( member(B,X3,A5)
          & ( aa(B,option(A),Mb,X3) = aa(A,option(A),some(A),Yb) ) ) ) ).

% ran_restrictD
tff(fact_6195_ran__def,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A))] : ran(B,A,Mb) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ach(fun(B,option(A)),fun(A,$o),Mb)) ).

% ran_def
tff(fact_6196_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),Xb: B,Yb: A,Z: A] :
      ( ( aa(B,option(A),Mb,Xb) = aa(A,option(A),some(A),Yb) )
     => ( inj_on(B,option(A),Mb,dom(B,A,Mb))
       => ( ~ member(A,Z,ran(B,A,Mb))
         => ( ran(B,A,fun_upd(B,option(A),Mb,Xb,aa(A,option(A),some(A),Z))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),ran(B,A,Mb)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_6197_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      num_of_nat(aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb),inc(num_of_nat(Nb)),one2) ).

% num_of_nat.simps(2)
tff(fact_6198_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B))] :
      ( ( dom(A,B,F3) = bot_bot(set(A)) )
    <=> ! [X4: A] : aa(A,option(B),F3,X4) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_6199_dom__const,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : dom(A,B,aTP_Lamp_aci(fun(A,B),fun(A,option(B)),F3)) = top_top(set(A)) ).

% dom_const
tff(fact_6200_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_ace(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_6201_finite__graph__iff__finite__dom,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),graph(A,B,Mb))
    <=> aa(set(A),$o,finite_finite2(A),dom(A,B,Mb)) ) ).

% finite_graph_iff_finite_dom
tff(fact_6202_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) ) ).

% add_neg_numeral_special(5)
tff(fact_6203_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Mb))) ) ).

% add_neg_numeral_special(6)
tff(fact_6204_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),Xb: A,Yb: option(B)] :
      dom(A,B,fun_upd(A,option(B),F3,Xb,Yb)) = $ite(Yb = none(B),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),dom(A,B,F3))) ).

% dom_fun_upd
tff(fact_6205_finite__ran,axiom,
    ! [B: $tType,A: $tType,P: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,P))
     => aa(set(B),$o,finite_finite2(B),ran(A,B,P)) ) ).

% finite_ran
tff(fact_6206_insert__dom,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),Xb: B,Yb: A] :
      ( ( aa(B,option(A),F3,Xb) = aa(A,option(A),some(A),Yb) )
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),dom(B,A,F3)) = dom(B,A,F3) ) ) ).

% insert_dom
tff(fact_6207_domD,axiom,
    ! [A: $tType,B: $tType,A2: A,Mb: fun(A,option(B))] :
      ( member(A,A2,dom(A,B,Mb))
     => ? [B4: B] : aa(A,option(B),Mb,A2) = aa(B,option(B),some(B),B4) ) ).

% domD
tff(fact_6208_domI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),Mb,A2) = aa(A,option(A),some(A),B2) )
     => member(B,A2,dom(B,A,Mb)) ) ).

% domI
tff(fact_6209_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,F3))
     => ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
       => ? [X3: A] : aa(A,option(B),F3,X3) = none(B) ) ) ).

% finite_map_freshness
tff(fact_6210_finite__set__of__finite__maps,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => aa(set(fun(A,option(B))),$o,finite_finite2(fun(A,option(B))),aa(fun(fun(A,option(B)),$o),set(fun(A,option(B))),collect(fun(A,option(B))),aa(set(B),fun(fun(A,option(B)),$o),aTP_Lamp_acj(set(A),fun(set(B),fun(fun(A,option(B)),$o)),A5),B5))) ) ) ).

% finite_set_of_finite_maps
tff(fact_6211_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xb: num] : aa(num,A,numeral_numeral(A),inc(Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) ) ).

% numeral_inc
tff(fact_6212_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),P2: fun(fun(A,option(B)),$o)] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,Mb))
     => ( aa(fun(A,option(B)),$o,P2,aTP_Lamp_ace(A,option(B)))
       => ( ! [K: A,V3: B,M: fun(A,option(B))] :
              ( aa(set(A),$o,finite_finite2(A),dom(A,B,M))
             => ( ~ member(A,K,dom(A,B,M))
               => ( aa(fun(A,option(B)),$o,P2,M)
                 => aa(fun(A,option(B)),$o,P2,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V3))) ) ) )
         => aa(fun(A,option(B)),$o,P2,Mb) ) ) ) ).

% finite_Map_induct
tff(fact_6213_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),Xb: A] :
      ( ( dom(A,B,F3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))) )
    <=> ? [V7: B] : F3 = fun_upd(A,option(B),aTP_Lamp_ace(A,option(B)),Xb,aa(B,option(B),some(B),V7)) ) ).

% dom_eq_singleton_conv
tff(fact_6214_finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( finite5375528669736107172at_top(A,A5) = principal(set(A),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),A5),bot_bot(set(set(A))))) ) ) ).

% finite_subsets_at_top_finite
tff(fact_6215_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_ack(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Max.eq_fold'
tff(fact_6216_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_6217_comp__the__Some,axiom,
    ! [A: $tType] : aa(fun(A,option(A)),fun(A,A),comp(option(A),A,A,the2(A)),some(A)) = id(A) ).

% comp_the_Some
tff(fact_6218_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,A5: set(A),P2: fun(set(A),$o)] :
      ( ! [X7: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X7)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),A5)
           => aa(set(A),$o,P2,X7) ) )
     => eventually(set(A),P2,finite5375528669736107172at_top(A,A5)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_6219_finite__subsets__at__top__neq__bot,axiom,
    ! [A: $tType,A5: set(A)] : finite5375528669736107172at_top(A,A5) != bot_bot(filter(set(A))) ).

% finite_subsets_at_top_neq_bot
tff(fact_6220_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_6221_eventually__finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( eventually(set(A),P2,finite5375528669736107172at_top(A,A5))
      <=> aa(set(A),$o,P2,A5) ) ) ).

% eventually_finite_subsets_at_top_finite
tff(fact_6222_option_Osel,axiom,
    ! [A: $tType,X2: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X2)) = X2 ).

% option.sel
tff(fact_6223_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_6224_option_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),Option: option(B)] :
      case_option(A,B,F1,F22,Option) = $ite(Option = none(B),F1,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ).

% option.case_eq_if
tff(fact_6225_Option_Othese__def,axiom,
    ! [A: $tType,A5: set(option(A))] : these(A,A5) = aa(set(option(A)),set(A),image2(option(A),A,the2(A)),aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_nt(set(option(A)),fun(option(A),$o),A5))) ).

% Option.these_def
tff(fact_6226_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_6227_option_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P2,case_option(A,B,F1,F22,Option))
    <=> ~ ( ( ( Option = none(B) )
            & ~ aa(A,$o,P2,F1) )
          | ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
            & ~ aa(A,$o,P2,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_6228_option_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P2,case_option(A,B,F1,F22,Option))
    <=> ( ( ( Option = none(B) )
         => aa(A,$o,P2,F1) )
        & ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
         => aa(A,$o,P2,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel
tff(fact_6229_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,P2: fun(set(A),$o),A5: set(A)] :
      ( eventually(set(A),P2,finite5375528669736107172at_top(A,A5))
    <=> ? [X9: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X9)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X9),A5)
          & ! [Y7: set(A)] :
              ( ( aa(set(A),$o,finite_finite2(A),Y7)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X9),Y7)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Y7),A5) )
             => aa(set(A),$o,P2,Y7) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_6230_graph__eq__to__snd__dom,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B))] : graph(A,B,Mb) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_acl(fun(A,option(B)),fun(A,product_prod(A,B)),Mb)),dom(A,B,Mb)) ).

% graph_eq_to_snd_dom
tff(fact_6231_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A5: set(A)] : finite5375528669736107172at_top(A,A5) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image2(set(A),filter(set(A)),aTP_Lamp_acn(set(A),fun(set(A),filter(set(A))),A5)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aco(set(A),fun(set(A),$o),A5)))) ).

% finite_subsets_at_top_def
tff(fact_6232_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,set(B)),A5: set(B),F4: filter(A)] :
      ( filterlim(A,set(B),F3,finite5375528669736107172at_top(B,A5),F4)
    <=> ! [X9: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),X9)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),A5) )
         => eventually(A,aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_acp(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),F3),A5),X9),F4) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_6233_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_acq(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Sup_fin.eq_fold'
tff(fact_6234_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_acr(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Inf_fin.eq_fold'
tff(fact_6235_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = A2 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_6236_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = Xb ) ).

% Sup_fin.singleton
tff(fact_6237_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = Xb ) ).

% Inf_fin.singleton
tff(fact_6238_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A2) = A2 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_6239_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_6240_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_6241_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_6242_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_6243_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A2) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_6244_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_6245_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A5) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_6246_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_6247_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5))
             => ! [A9: A] :
                  ( member(A,A9,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A9) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_6248_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_6249_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),Xb)
             => ! [A9: A] :
                  ( member(A,A9,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),Xb) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_6250_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xb) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),Xb) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_6251_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5))
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X4) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_6252_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),Xb)
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xb) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_6253_cSup__eq__Sup__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = aa(set(A),A,lattic5882676163264333800up_fin(A),X5) ) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_6254_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_6255_cInf__eq__Inf__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X5) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_6256_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_6257_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_6258_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_6259_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_6260_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_6261_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X3)),aa(A,A,H,Y))
         => ( aa(set(A),$o,finite_finite2(A),N3)
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic7752659483105999362nf_fin(A),N3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),image2(A,A,H),N3)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_6262_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X3)),aa(A,A,H,Y))
         => ( aa(set(A),$o,finite_finite2(A),N3)
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic5882676163264333800up_fin(A),N3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),image2(A,A,H),N3)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_6263_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_6264_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A5) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_6265_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_6266_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A5),A5) ) ) ) ) ).

% Inf_fin.closed
tff(fact_6267_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A5),A5) ) ) ) ) ).

% Sup_fin.closed
tff(fact_6268_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_6269_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != 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)),A5),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_6270_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != 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)),A5),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_6271_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = finite_fold(A,A,inf_inf(A),Xb,A5) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_6272_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = finite_fold(A,A,sup_sup(A),Xb,A5) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_6273_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_acs(set(A),fun(A,fun(A,$o)),A5),Xb))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_6274_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_act(set(A),fun(set(A),fun(A,$o)),A5),B5))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_6275_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_acu(set(A),fun(A,fun(A,$o)),A5),Xb))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_6276_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_acv(set(A),fun(set(A),fun(A,$o)),A5),B5))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_6277_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = $ite(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)),insert(A),Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(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)),insert(A),Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_6278_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = $ite(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)),insert(A),Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(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)),insert(A),Xb),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_6279_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = $ite(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)),insert(A),Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(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)),insert(A),Xb),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_6280_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = $ite(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)),insert(A),Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(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)),insert(A),Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_6281_pos__deriv__imp__strict__mono,axiom,
    ! [F3: fun(real,real),F9: fun(real,real)] :
      ( ! [X3: real] : has_field_derivative(real,F3,aa(real,real,F9,X3),topolo174197925503356063within(real,X3,top_top(set(real))))
     => ( ! [X3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F9,X3))
       => order_strict_mono(real,real,F3) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_6282_sorted__key__list__of__set__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B)] : linord144544945434240204of_set(A,B,F3) = finite_folding_F(A,list(A),linorder_insort_key(A,B,F3),nil(A)) ) ).

% sorted_key_list_of_set_def
tff(fact_6283_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_strict_mono(A,B,F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb) ) ) ) ).

% strict_mono_less_eq
tff(fact_6284_strict__mono__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [R2: fun(A,B),Mb: A,Nb: A] :
          ( order_strict_mono(A,B,R2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),Nb)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,R2,Mb)),aa(A,B,R2,Nb)) ) ) ) ).

% strict_mono_leD
tff(fact_6285_folding__on_OF_Ocong,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z: B] : finite_folding_F(A,B,F3,Z) = finite_folding_F(A,B,F3,Z) ).

% folding_on.F.cong
tff(fact_6286_strict__mono__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_strict_mono(A,B,F3)
         => ( ( aa(A,B,F3,Xb) = aa(A,B,F3,Yb) )
          <=> ( Xb = Yb ) ) ) ) ).

% strict_mono_eq
tff(fact_6287_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_strict_mono(A,B,F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb) ) ) ) ).

% strict_mono_less
tff(fact_6288_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_strict_mono(A,B,F3)
        <=> ! [X4: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),aa(A,B,F3,Y4)) ) ) ) ).

% strict_mono_def
tff(fact_6289_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y)) )
         => order_strict_mono(A,B,F3) ) ) ).

% strict_monoI
tff(fact_6290_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_strict_mono(A,B,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb)) ) ) ) ).

% strict_monoD
tff(fact_6291_strict__mono__add,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A] : order_strict_mono(A,A,aTP_Lamp_ls(A,fun(A,A),K2)) ) ).

% strict_mono_add
tff(fact_6292_strict__mono__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_strict_mono(A,B,F3)
         => order_mono(A,B,F3) ) ) ).

% strict_mono_mono
tff(fact_6293_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xb),aa(set(A),B,finite_folding_F(A,B,F3,Z),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)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_6294_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),Xb: A,Z: B] :
      ( finite_folding_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),A5) = aa(B,B,aa(A,fun(B,B),F3,Xb),aa(set(A),B,finite_folding_F(A,B,F3,Z),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)),insert(A),Xb),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_6295_strict__mono__imp__increasing,axiom,
    ! [F3: fun(nat,nat),Nb: nat] :
      ( order_strict_mono(nat,nat,F3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,F3,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_6296_infinite__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => ? [R3: fun(nat,nat)] :
          ( order_strict_mono(nat,nat,R3)
          & ! [N4: nat] : member(nat,aa(nat,nat,R3,N4),S2) ) ) ).

% infinite_enumerate
tff(fact_6297_folding__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,S2)
         => ( member(A,Y,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y)),aa(A,fun(B,B),F3,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y)) ) ) )
     => finite_folding_on(A,B,S2,F3) ) ).

% folding_on.intro
tff(fact_6298_folding__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,Yb: A] :
      ( finite_folding_on(A,B,S2,F3)
     => ( member(A,Xb,S2)
       => ( member(A,Yb,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Yb)),aa(A,fun(B,B),F3,Xb)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Xb)),aa(A,fun(B,B),F3,Yb)) ) ) ) ) ).

% folding_on.comp_fun_commute_on
tff(fact_6299_folding__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S2,F3)
    <=> ! [X4: A,Y4: A] :
          ( member(A,X4,S2)
         => ( member(A,Y4,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y4)),aa(A,fun(B,B),F3,X4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X4)),aa(A,fun(B,B),F3,Y4)) ) ) ) ) ).

% folding_on_def
tff(fact_6300_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_strict_mono(nat,A,F3)
        <=> ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N))) ) ) ).

% strict_mono_Suc_iff
tff(fact_6301_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_abp(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_6302_folding__on_Oempty,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Z: B] :
      ( finite_folding_on(A,B,S2,F3)
     => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),bot_bot(set(A))) = Z ) ) ).

% folding_on.empty
tff(fact_6303_folding__on_Oinfinite,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F3)
     => ( ~ aa(set(A),$o,finite_finite2(A),A5)
       => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),A5) = Z ) ) ) ).

% folding_on.infinite
tff(fact_6304_folding__on_Oeq__fold,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Z: B,A5: set(A)] :
      ( finite_folding_on(A,B,S2,F3)
     => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),A5) = finite_fold(A,B,F3,Z,A5) ) ) ).

% folding_on.eq_fold
tff(fact_6305_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_mr(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_6306_card__def,axiom,
    ! [A: $tType] : finite_card(A) = finite_folding_F(A,nat,aTP_Lamp_abp(A,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_6307_summable__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G3: fun(nat,nat),F3: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G3)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,aa(set(nat),set(nat),image2(nat,nat,G3),top_top(set(nat))))
               => ( aa(nat,A,F3,N2) = zero_zero(A) ) )
           => ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_acw(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G3),F3))
            <=> summable(A,F3) ) ) ) ) ).

% summable_mono_reindex
tff(fact_6308_sums__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G3: fun(nat,nat),F3: fun(nat,A),C3: A] :
          ( order_strict_mono(nat,nat,G3)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,aa(set(nat),set(nat),image2(nat,nat,G3),top_top(set(nat))))
               => ( aa(nat,A,F3,N2) = zero_zero(A) ) )
           => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_acw(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G3),F3),C3)
            <=> sums(A,F3,C3) ) ) ) ) ).

% sums_mono_reindex
tff(fact_6309_suminf__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [G3: fun(nat,nat),F3: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G3)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,aa(set(nat),set(nat),image2(nat,nat,G3),top_top(set(nat))))
               => ( aa(nat,A,F3,N2) = zero_zero(A) ) )
           => ( suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_acx(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G3),F3)) = suminf(A,F3) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_6310_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G3: fun(nat,nat)] :
          ( ! [X3: nat,Y: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Y)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,X3))),real_V7770717601297561774m_norm(A,aa(nat,A,F3,Y))) )
         => ( order_strict_mono(nat,nat,G3)
           => ( bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_acy(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F3),G3),at_top(nat))
            <=> bfun(nat,A,F3,at_top(nat)) ) ) ) ) ).

% increasing_Bseq_subseq_iff
tff(fact_6311_folding__on_Oinsert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xb),aa(set(A),B,finite_folding_F(A,B,F3,Z),A5)) ) ) ) ) ) ).

% folding_on.insert
tff(fact_6312_folding__idem__on_Oinsert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite1890593828518410140dem_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xb),aa(set(A),B,finite_folding_F(A,B,F3,Z),A5)) ) ) ) ) ).

% folding_idem_on.insert_idem
tff(fact_6313_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xb),finite_fold(A,B,F3,Z,A5)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_6314_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( member(A,Xb,S2)
       => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Xb)),aa(A,fun(B,B),F3,Xb)) = aa(A,fun(B,B),F3,Xb) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_6315_folding__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,Yb: A] :
      ( finite1890593828518410140dem_on(A,B,S2,F3)
     => ( member(A,Xb,S2)
       => ( member(A,Yb,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Xb)),aa(A,fun(B,B),F3,Xb)) = aa(A,fun(B,B),F3,Xb) ) ) ) ) ).

% folding_idem_on.comp_fun_idem_on
tff(fact_6316_comp__fun__idem__on_Ofun__left__idem,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( member(A,Xb,S2)
       => ( aa(B,B,aa(A,fun(B,B),F3,Xb),aa(B,B,aa(A,fun(B,B),F3,Xb),Z)) = aa(B,B,aa(A,fun(B,B),F3,Xb),Z) ) ) ) ).

% comp_fun_idem_on.fun_left_idem
tff(fact_6317_comp__fun__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => finite4664212375090638736ute_on(A,B,S2,F3) ) ).

% comp_fun_idem_on.axioms(1)
tff(fact_6318_folding__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F3)
     => finite_folding_on(A,B,S2,F3) ) ).

% folding_idem_on.axioms(1)
tff(fact_6319_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F3: fun(A,fun(B,B)),G3: fun(C,A),R: set(C)] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G3),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,F3),G3)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_6320_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,Xb),Z),A5) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_6321_folding__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F3)
    <=> ( finite_folding_on(A,B,S2,F3)
        & finite6916993218817215295axioms(A,B,S2,F3) ) ) ).

% folding_idem_on_def
tff(fact_6322_folding__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S2,F3)
     => ( finite6916993218817215295axioms(A,B,S2,F3)
       => finite1890593828518410140dem_on(A,B,S2,F3) ) ) ).

% folding_idem_on.intro
tff(fact_6323_folding__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite6916993218817215295axioms(A,B,S2,F3)
    <=> ! [X4: A,Y4: A] :
          ( member(A,X4,S2)
         => ( member(A,Y4,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X4)),aa(A,fun(B,B),F3,X4)) = aa(A,fun(B,B),F3,X4) ) ) ) ) ).

% folding_idem_on_axioms_def
tff(fact_6324_folding__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,S2)
         => ( member(A,Y,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,X3)) = aa(A,fun(B,B),F3,X3) ) ) )
     => finite6916993218817215295axioms(A,B,S2,F3) ) ).

% folding_idem_on_axioms.intro
tff(fact_6325_folding__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F3)
     => finite6916993218817215295axioms(A,B,S2,F3) ) ).

% folding_idem_on.axioms(2)
tff(fact_6326_positive__rat,axiom,
    ! [A2: int,B2: int] :
      ( aa(rat,$o,positive,fract(A2,B2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ).

% positive_rat
tff(fact_6327_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(set(nat),set(A),image2(nat,A,nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),take(A,L,Xs)) ) ) ).

% nth_image
tff(fact_6328_take__eq__Nil2,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,Nb,Xs) )
    <=> ( ( Nb = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_6329_take__eq__Nil,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( take(A,Nb,Xs) = nil(A) )
    <=> ( ( Nb = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_6330_take0,axiom,
    ! [A: $tType,X: list(A)] : take(A,zero_zero(nat),X) = nil(A) ).

% take0
tff(fact_6331_take__all,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)
     => ( take(A,Nb,Xs) = Xs ) ) ).

% take_all
tff(fact_6332_take__all__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( take(A,Nb,Xs) = Xs )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% take_all_iff
tff(fact_6333_nth__take,axiom,
    ! [A: $tType,I2: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
     => ( aa(nat,A,nth(A,take(A,Nb,Xs)),I2) = aa(nat,A,nth(A,Xs),I2) ) ) ).

% nth_take
tff(fact_6334_take__update__cancel,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A),Yb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( take(A,Nb,list_update(A,Xs,Mb,Yb)) = take(A,Nb,Xs) ) ) ).

% take_update_cancel
tff(fact_6335_nths__upt__eq__take,axiom,
    ! [A: $tType,L: list(A),Nb: nat] : nths(A,L,aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = take(A,Nb,L) ).

% nths_upt_eq_take
tff(fact_6336_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Xs: list(A),Ys: list(B)] : dom(A,B,map_upds(A,B,Mb,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))),dom(A,B,Mb)) ).

% dom_map_upds
tff(fact_6337_set__take__subset,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Nb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_take_subset
tff(fact_6338_less__rat__def,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xb),Yb)
    <=> aa(rat,$o,positive,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Yb),Xb)) ) ).

% less_rat_def
tff(fact_6339_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_6340_take__Nil,axiom,
    ! [A: $tType,Nb: nat] : take(A,Nb,nil(A)) = nil(A) ).

% take_Nil
tff(fact_6341_take__update__swap,axiom,
    ! [A: $tType,Mb: nat,Xs: list(A),Nb: nat,Xb: A] : take(A,Mb,list_update(A,Xs,Nb,Xb)) = list_update(A,take(A,Mb,Xs),Nb,Xb) ).

% take_update_swap
tff(fact_6342_take__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ! [I3: nat] : take(A,I3,Xs) = take(A,I3,Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_6343_distinct__take,axiom,
    ! [A: $tType,Xs: list(A),I2: nat] :
      ( distinct(A,Xs)
     => distinct(A,take(A,I2,Xs)) ) ).

% distinct_take
tff(fact_6344_in__set__takeD,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),take(A,Nb,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_takeD
tff(fact_6345_set__take__subset__set__take,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Mb,Xs))),aa(list(A),set(A),set2(A),take(A,Nb,Xs))) ) ).

% set_take_subset_set_take
tff(fact_6346_nth__take__lemma,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Ys))
       => ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K2)
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) )
         => ( take(A,K2,Xs) = take(A,K2,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_6347_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),Xb: A,Yb: B,Xs: list(A),Ys: list(B)] :
      map_upds(A,B,fun_upd(A,option(B),F3,Xb,aa(B,option(B),some(B),Yb)),Xs,Ys) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))),map_upds(A,B,F3,Xs,Ys),fun_upd(A,option(B),map_upds(A,B,F3,Xs,Ys),Xb,aa(B,option(B),some(B),Yb))) ).

% map_upd_upds_conv_if
tff(fact_6348_Rat_Opositive_Orep__eq,axiom,
    ! [Xb: rat] :
      ( aa(rat,$o,positive,Xb)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb)))) ) ).

% Rat.positive.rep_eq
tff(fact_6349_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2))
     => ~ ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Ys))
             => ( ( take(A,I3,Xs) = take(A,I3,Ys) )
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Ys),I3)),R2) ) ) ) ) ).

% lex_take_index
tff(fact_6350_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys),lex(A,R2)) ).

% Nil_notin_lex
tff(fact_6351_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)),lex(A,R2)) ).

% Nil2_notin_lex
tff(fact_6352_Rat_Opositive__def,axiom,
    positive = aa(fun(product_prod(int,int),$o),fun(rat,$o),map_fun(rat,product_prod(int,int),$o,$o,rep_Rat,id($o)),aTP_Lamp_acz(product_prod(int,int),$o)) ).

% Rat.positive_def
tff(fact_6353_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
    <=> ? [Y4: A,N: nat] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N)),Y4),R2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
          & ( Ys = list_update(A,Xs,N,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_6354_not__Nil__listrel1,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xs),listrel1(A,R2)) ).

% not_Nil_listrel1
tff(fact_6355_not__listrel1__Nil,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)),listrel1(A,R2)) ).

% not_listrel1_Nil
tff(fact_6356_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_6357_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,Xb: list(A),Yb: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Yb),transitive_rtrancl(list(A),listrel1(A,R2)))
     => ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Yb) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_6358_listrel1__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S)
     => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel1(A,S)) ) ).

% listrel1_mono
tff(fact_6359_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,transitive_rtrancl(A,R2))),transitive_rtrancl(list(A),listrel1(A,R2))) ).

% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_6360_listrel1p__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( listrel1p(A,R2,Xs,Ys)
    <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).

% listrel1p_def
tff(fact_6361_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_ada(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lenlex_conv
tff(fact_6362_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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_6363_lenlex__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ 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)
     => ~ 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_6364_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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_6365_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_6366_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_ts(A,fun(A,$o))) = ord_max(A) ) ) ).

% dual_min
tff(fact_6367_rp__inv__image__rp,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),F3: fun(B,A)] :
      ( fun_reduction_pair(A,P2)
     => fun_reduction_pair(B,aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),fun_rp_inv_image(A,B),P2),F3)) ) ).

% rp_inv_image_rp
tff(fact_6368_ord_Omin__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),A2: A,B2: A] :
      aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2),A2,B2) ).

% ord.min_def
tff(fact_6369_ord_Omin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : min(A,Less_eq) = min(A,Less_eq) ).

% ord.min.cong
tff(fact_6370_rp__inv__image__def,axiom,
    ! [B: $tType,A: $tType] : fun_rp_inv_image(A,B) = aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_adb(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))))) ).

% rp_inv_image_def
tff(fact_6371_card__Min__le__sum,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image2(A,nat,F3),A5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)) ) ).

% card_Min_le_sum
tff(fact_6372_in__inv__image,axiom,
    ! [A: $tType,B: $tType,Xb: A,Yb: A,R2: set(product_prod(B,B)),F3: fun(A,B)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),inv_image(B,A,R2,F3))
    <=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb)),R2) ) ).

% in_inv_image
tff(fact_6373_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = Xb ) ).

% Min_singleton
tff(fact_6374_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X4) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6375_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
            <=> ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X4) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_6376_Min__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A5: set(A),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_qj(B,fun(A,B),C3)),A5)) = C3 ) ) ) ) ).

% Min_const
tff(fact_6377_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,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_6378_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,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_6379_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_6380_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5),A5) ) ) ) ).

% Min_in
tff(fact_6381_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xb) ) ) ) ).

% Min_le
tff(fact_6382_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [Y: A] :
                ( member(A,Y,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) )
           => ( member(A,Xb,A5)
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = Xb ) ) ) ) ) ).

% Min_eqI
tff(fact_6383_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),A2) ) ) ) ).

% Min.coboundedI
tff(fact_6384_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = Mb )
            <=> ( member(A,Mb,A5)
                & ! [X4: A] :
                    ( member(A,X4,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X4) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6385_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xb)
            <=> ? [X4: A] :
                  ( member(A,X4,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xb) ) ) ) ) ) ).

% Min_le_iff
tff(fact_6386_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798350308766er_Min(A),A5) )
            <=> ( member(A,Mb,A5)
                & ! [X4: A] :
                    ( member(A,X4,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X4) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6387_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
             => ! [A9: A] :
                  ( member(A,A9,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A9) ) ) ) ) ) ).

% Min.boundedE
tff(fact_6388_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min.boundedI
tff(fact_6389_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xb)
            <=> ? [X4: A] :
                  ( member(A,X4,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xb) ) ) ) ) ) ).

% Min_less_iff
tff(fact_6390_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [B4: A] :
                ( member(A,B4,A5)
               => aa(A,$o,aa(A,fun(A,$o),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)),insert(A),A2),A5)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_6391_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = aa(set(A),A,lattic643756798350308766er_Min(A),X5) ) ) ) ) ).

% cInf_eq_Min
tff(fact_6392_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ) ) ) ).

% Min_Inf
tff(fact_6393_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_6394_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M4: set(A),N3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M4),N3)
         => ( ( M4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N3)),aa(set(A),A,lattic643756798350308766er_Min(A),M4)) ) ) ) ) ).

% Min_antimono
tff(fact_6395_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min.subset_imp
tff(fact_6396_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,B)),F3: fun(A,B)] : inv_image(B,A,R2,F3) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_adc(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),R2),F3))) ).

% inv_image_def
tff(fact_6397_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(A,B,F3,aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F3),A5)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_6398_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S2: set(A),F3: fun(A,B),K2: B] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_rv(fun(A,B),fun(B,fun(A,B)),F3),K2)),S2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F3),S2))),K2) ) ) ) ) ).

% Min_add_commute
tff(fact_6399_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_ts(A,fun(A,$o))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_6400_f__arg__min__list__f,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( aa(A,B,F3,arg_min_list(A,B,F3,Xs)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F3),aa(list(A),set(A),set2(A),Xs))) ) ) ) ).

% f_arg_min_list_f
tff(fact_6401_linorder_OMax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : lattices_Max(A,Less_eq) = lattices_Max(A,Less_eq) ).

% linorder.Max.cong
tff(fact_6402_arg__min__list__in,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => member(A,arg_min_list(A,B,F3,Xs),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% arg_min_list_in
tff(fact_6403_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_add(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Min.eq_fold'
tff(fact_6404_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_6405_min__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)) ).

% min_Suc_Suc
tff(fact_6406_min__0R,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_6407_min__0L,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),Nb) = zero_zero(nat) ).

% min_0L
tff(fact_6408_min_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ).

% min.right_idem
tff(fact_6409_min_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ).

% min.left_idem
tff(fact_6410_min_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A2),A2) = A2 ) ).

% min.idem
tff(fact_6411_take__take,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] : take(A,Nb,take(A,Mb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),Mb),Xs) ).

% take_take
tff(fact_6412_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% min.bounded_iff
tff(fact_6413_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_6414_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_6415_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Yb) ) ) ) ).

% min_less_iff_conj
tff(fact_6416_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_6417_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb3
tff(fact_6418_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),Xb) = Xb ) ).

% min_top
tff(fact_6419_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),top_top(A)) = Xb ) ).

% min_top2
tff(fact_6420_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_6421_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),Xb) = bot_bot(A) ) ).

% min_bot
tff(fact_6422_length__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),take(A,Nb,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_take
tff(fact_6423_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)) = Xb ) ).

% max_min_same(1)
tff(fact_6424_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)),Xb) = Xb ) ).

% max_min_same(2)
tff(fact_6425_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)),Yb) = Yb ) ).

% max_min_same(3)
tff(fact_6426_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Yb: A,Xb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Yb),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)) = Yb ) ).

% max_min_same(4)
tff(fact_6427_take__replicate,axiom,
    ! [A: $tType,I2: nat,K2: nat,Xb: A] : take(A,I2,replicate(A,K2,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I2),K2),Xb) ).

% take_replicate
tff(fact_6428_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(1)
tff(fact_6429_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xb)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_6430_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xb)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_6431_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_6432_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_6433_Int__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),A2)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)) ) ).

% Int_atMost
tff(fact_6434_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(2)
tff(fact_6435_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(3)
tff(fact_6436_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(4)
tff(fact_6437_Int__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMost
tff(fact_6438_Int__atLeastAtMostR1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),B2)),set_or1337092689740270186AtMost(A,C3,D3)) = set_or1337092689740270186AtMost(A,C3,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMostR1
tff(fact_6439_Int__atLeastAtMostL1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(A,set(A),set_ord_atMost(A),D3)) = set_or1337092689740270186AtMost(A,A2,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastAtMostL1
tff(fact_6440_Int__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_atLeastLessThan
tff(fact_6441_Int__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C3,D3)) = set_or5935395276787703475ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_greaterThanLessThan
tff(fact_6442_Int__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: 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,C3,D3)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D3)) ) ).

% Int_greaterThanAtMost
tff(fact_6443_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min_insert
tff(fact_6444_Min_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(A),A,lattic643756798350308766er_Min(A),A5) ) ) ) ) ).

% Min.in_idem
tff(fact_6445_min__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),Mb: A,Nb: A] :
          ( order_mono(A,B,F3)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F3,Mb)),aa(A,B,F3,Nb)) = aa(A,B,F3,aa(A,A,aa(A,fun(A,A),ord_min(A),Mb),Nb)) ) ) ) ).

% min_of_mono
tff(fact_6446_max__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_antimono(A,B,F3)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb)) = aa(A,B,F3,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)) ) ) ) ).

% max_of_antimono
tff(fact_6447_min__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),Xb: A,Yb: A] :
          ( order_antimono(A,B,F3)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb)) = aa(A,B,F3,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb)) ) ) ) ).

% min_of_antimono
tff(fact_6448_nat__mult__min__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)) ).

% nat_mult_min_right
tff(fact_6449_nat__mult__min__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ).

% nat_mult_min_left
tff(fact_6450_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xb: A,Yb: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Yb)) ) ).

% minus_max_eq_min
tff(fact_6451_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xb: A,Yb: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Yb)) ) ).

% minus_min_eq_max
tff(fact_6452_inf__min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( inf_inf(A) = ord_min(A) ) ) ).

% inf_min
tff(fact_6453_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.strict_coboundedI2
tff(fact_6454_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.strict_coboundedI1
tff(fact_6455_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_6456_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% min.strict_boundedE
tff(fact_6457_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_6458_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Yb),Z)) ) ).

% min_add_distrib_left
tff(fact_6459_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),ord_min(A),Yb),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Yb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ).

% min_add_distrib_right
tff(fact_6460_min_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),B2),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),C3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) ) ).

% min.left_commute
tff(fact_6461_min_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: 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),B2),A2) ) ).

% min.commute
tff(fact_6462_min_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) ) ).

% min.assoc
tff(fact_6463_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Yb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,semiring_1_of_nat(A),Yb)) ) ).

% of_nat_min
tff(fact_6464_min__max__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),ord_max(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),A2),C3)) ) ).

% min_max_distrib2
tff(fact_6465_min__max__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),A2)) ) ).

% min_max_distrib1
tff(fact_6466_max__min__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C3)) ) ).

% max_min_distrib2
tff(fact_6467_max__min__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)),A2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A2)),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),A2)) ) ).

% max_min_distrib1
tff(fact_6468_of__int__min,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,Yb: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_min(int),Xb),Yb)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(int,A,ring_1_of_int(A),Xb)),aa(int,A,ring_1_of_int(A),Yb)) ) ).

% of_int_min
tff(fact_6469_greaterThan__Int__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),A2)),aa(A,set(A),set_ord_lessThan(A),B2)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)) ) ).

% greaterThan_Int_greaterThan
tff(fact_6470_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Yb: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Yb),Z)) ) ).

% min_diff_distrib_left
tff(fact_6471_min__diff,axiom,
    ! [Mb: nat,I2: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),I2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),I2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),I2) ).

% min_diff
tff(fact_6472_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_6473_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.coboundedI2
tff(fact_6474_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.coboundedI1
tff(fact_6475_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_6476_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_6477_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2) ) ).

% min.cobounded2
tff(fact_6478_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2) ) ).

% min.cobounded1
tff(fact_6479_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_6480_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) ) ) ) ).

% min.boundedI
tff(fact_6481_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% min.boundedE
tff(fact_6482_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% min.orderI
tff(fact_6483_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.orderE
tff(fact_6484_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),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),C3),D3)) ) ) ) ).

% min.mono
tff(fact_6485_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Yb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Yb),Xb)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb) = Yb ) ) ) ).

% min_absorb2
tff(fact_6486_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Yb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb) = Xb ) ) ) ).

% min_absorb1
tff(fact_6487_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),A2,B2) ) ).

% min_def
tff(fact_6488_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),X),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa),X,Xa) ) ).

% min_def_raw
tff(fact_6489_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: A,Xb: A,Yb: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P),Yb)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P),Yb))) ) ).

% max_mult_distrib_left
tff(fact_6490_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: A,Xb: A,Yb: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P),Yb)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P),Yb))) ) ).

% min_mult_distrib_left
tff(fact_6491_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A,P: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb)),P) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),P)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),P))) ) ).

% max_mult_distrib_right
tff(fact_6492_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Yb: A,P: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb)),P) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),P)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P)),aa(A,A,aa(A,fun(A,A),times_times(A),Yb),P))) ) ).

% min_mult_distrib_right
tff(fact_6493_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A,P: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Yb),P) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xb,P)),divide_divide(A,Yb,P)),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xb,P)),divide_divide(A,Yb,P))) ) ).

% max_divide_distrib_right
tff(fact_6494_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Yb: A,P: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Yb),P) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xb,P)),divide_divide(A,Yb,P)),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xb,P)),divide_divide(A,Yb,P))) ) ).

% min_divide_distrib_right
tff(fact_6495_min__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),Mb) = case_nat(nat,zero_zero(nat),aTP_Lamp_ade(nat,fun(nat,nat),Nb),Mb) ).

% min_Suc1
tff(fact_6496_min__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_adf(nat,fun(nat,nat),Nb),Mb) ).

% min_Suc2
tff(fact_6497_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),S2)) = $ite(S2 = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,complete_Inf_Inf(A),S2))) ) ) ) ).

% Inf_insert_finite
tff(fact_6498_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X3)),aa(A,A,H,Y))
         => ( aa(set(A),$o,finite_finite2(A),N3)
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798350308766er_Min(A),N3)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,H),N3)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_6499_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(A),A,lattic643756798350308766er_Min(A),A5) ) ) ) ) ) ).

% Min.subset
tff(fact_6500_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5),A5) ) ) ) ) ).

% Min.closed
tff(fact_6501_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_6502_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != 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)),A5),B5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),aa(set(A),A,lattic643756798350308766er_Min(A),B5)) ) ) ) ) ) ) ).

% Min.union
tff(fact_6503_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = finite_fold(A,A,ord_min(A),Xb,A5) ) ) ) ).

% Min.eq_fold
tff(fact_6504_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = $ite(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)),insert(A),Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(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)),insert(A),Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Min.remove
tff(fact_6505_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)) = $ite(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)),insert(A),Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(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)),insert(A),Xb),bot_bot(set(A))))))) ) ) ) ).

% Min.insert_remove
tff(fact_6506_lexord__take__index__conv,axiom,
    ! [A: $tType,Xb: list(A),Yb: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Yb),lexord(A,R2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xb)),aa(list(A),nat,size_size(list(A)),Yb))
          & ( take(A,aa(list(A),nat,size_size(list(A)),Xb),Yb) = Xb ) )
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xb)),aa(list(A),nat,size_size(list(A)),Yb)))
            & ( take(A,I4,Xb) = take(A,I4,Yb) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xb),I4)),aa(nat,A,nth(A,Yb),I4)),R2) ) ) ) ).

% lexord_take_index_conv
tff(fact_6507_lenlex__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R2)),aTP_Lamp_adg(list(A),product_prod(nat,list(A)))) ).

% lenlex_def
tff(fact_6508_less__than__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Yb),less_than)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Yb) ) ).

% less_than_iff
tff(fact_6509_inf__int__def,axiom,
    inf_inf(int) = ord_min(int) ).

% inf_int_def
tff(fact_6510_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_6511_lexord__linear,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: list(A),Yb: list(A)] :
      ( ! [A4: A,B4: A] :
          ( 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 )
          | 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) )
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Yb),lexord(A,R2))
        | ( Xb = Yb )
        | 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)),Yb),Xb),lexord(A,R2)) ) ) ).

% lexord_linear
tff(fact_6512_lexord__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ 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)
     => ~ 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_6513_lexord__Nil__right,axiom,
    ! [A: $tType,Xb: list(A),R2: set(product_prod(A,A))] : ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),nil(A)),lexord(A,R2)) ).

% lexord_Nil_right
tff(fact_6514_pair__less__def,axiom,
    fun_pair_less = lex_prod(nat,nat,less_than,less_than) ).

% pair_less_def
tff(fact_6515_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A,Y: A,Z2: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),R2)
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
             => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z2),R2) ) ) )
     => ( 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))
       => ( 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))
         => 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_6516_lexord__lex,axiom,
    ! [A: $tType,Xb: list(A),Yb: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Yb),lex(A,R2))
    <=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Yb),lexord(A,R2))
        & ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Yb) ) ) ) ).

% lexord_lex
tff(fact_6517_mlex__prod__def,axiom,
    ! [A: $tType,F3: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F3,R) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R),aTP_Lamp_adh(fun(A,nat),fun(A,product_prod(nat,A)),F3)) ).

% mlex_prod_def
tff(fact_6518_List_Olexordp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( lexordp(A,R2,Xs,Ys)
    <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2)))) ) ).

% List.lexordp_def
tff(fact_6519_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_adi(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys)) ).

% set_zip
tff(fact_6520_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_6521_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_6522_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_6523_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I2: nat,Xb: A,J: nat,Yb: B] : zip(A,B,replicate(A,I2,Xb),replicate(B,J,Yb)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I2),J),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb)) ).

% zip_replicate
tff(fact_6524_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_6525_nth__zip,axiom,
    ! [A: $tType,B: $tType,I2: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_6526_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,Xb: A,Ys: list(B),Yb: B] : zip(A,B,list_update(A,Xs,I2,Xb),list_update(B,Ys,I2,Yb)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb)) ).

% zip_update
tff(fact_6527_take__zip,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] : take(product_prod(A,B),Nb,zip(A,B,Xs,Ys)) = zip(A,B,take(A,Nb,Xs),take(B,Nb,Ys)) ).

% take_zip
tff(fact_6528_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_6529_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_6530_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_6531_zip__same,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs)))
    <=> ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_6532_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,Xb: A,Yb: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => ~ ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ~ member(B,Yb,aa(list(B),set(B),set2(B),Ys)) ) ) ).

% in_set_zipE
tff(fact_6533_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Yb: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_zip_leftD
tff(fact_6534_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,Xb: A,Yb: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => member(B,Yb,aa(list(B),set(B),set2(B),Ys)) ) ).

% set_zip_rightD
tff(fact_6535_update__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),I2: nat,Xy: product_prod(A,B)] : list_update(product_prod(A,B),zip(A,B,Xs,Ys),I2,Xy) = zip(A,B,list_update(A,Xs,I2,aa(product_prod(A,B),A,product_fst(A,B),Xy)),list_update(B,Ys,I2,aa(product_prod(A,B),B,product_snd(A,B),Xy))) ).

% update_zip
tff(fact_6536_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P2: fun(list(product_prod(A,B)),$o)] :
      ( ! [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 = take(A,N2,Xs) )
             => ( ( Ws = take(B,N2,Ys) )
               => aa(list(product_prod(A,B)),$o,P2,zip(A,B,Zs2,Ws)) ) ) ) )
     => aa(list(product_prod(A,B)),$o,P2,zip(A,B,Xs,Ys)) ) ).

% zip_obtain_same_length
tff(fact_6537_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X4: product_prod(A,A)] :
            ( member(product_prod(A,A),X4,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys)))
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),fequal(A)),X4) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_6538_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ~ ! [Y: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_6539_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Yb: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(B,Yb,aa(list(B),set(B),set2(B),Ys))
       => ~ ! [X3: A] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Yb),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_6540_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ! [X3: product_prod(list(A),list(A))] :
              ( member(product_prod(list(A),list(A)),X3,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys)))
             => aa(product_prod(list(A),list(A)),$o,aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_adj(list(A),fun(list(A),$o))),X3) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_6541_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ! [X3: product_prod(list(A),list(A))] :
          ( member(product_prod(list(A),list(A)),X3,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys)))
         => aa(product_prod(list(A),list(A)),$o,aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_adj(list(A),fun(list(A),$o))),X3) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ( concat(A,Xs) = concat(A,Ys) )
        <=> ( Xs = Ys ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_6542_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P: product_prod(A,B),Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),P,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
    <=> ? [N: nat] :
          ( ( aa(nat,A,nth(A,Xs),N) = aa(product_prod(A,B),A,product_fst(A,B),P) )
          & ( aa(nat,B,nth(B,Ys),N) = aa(product_prod(A,B),B,product_snd(A,B),P) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Ys)) ) ) ).

% in_set_zip
tff(fact_6543_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X4: product_prod(A,B)] :
            ( member(product_prod(A,B),X4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
           => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X4) ) ) ) ).

% listrel_iff_zip
tff(fact_6544_map__upds__fold__map__upd,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,Mb,Ks,Vs) = foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_adl(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Mb,zip(A,B,Ks,Vs)) ).

% map_upds_fold_map_upd
tff(fact_6545_listrel__rtrancl__refl,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),listrel(A,A,transitive_rtrancl(A,R2))) ).

% listrel_rtrancl_refl
tff(fact_6546_foldl__Nil,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),A2: A] : foldl(A,B,F3,A2,nil(B)) = A2 ).

% foldl_Nil
tff(fact_6547_foldl__cong,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: A,L: list(B),K2: list(B),F3: fun(A,fun(B,A)),G3: fun(A,fun(B,A))] :
      ( ( A2 = B2 )
     => ( ( L = K2 )
       => ( ! [A4: A,X3: B] :
              ( member(B,X3,aa(list(B),set(B),set2(B),L))
             => ( aa(B,A,aa(A,fun(B,A),F3,A4),X3) = aa(B,A,aa(A,fun(B,A),G3,A4),X3) ) )
         => ( foldl(A,B,F3,A2,L) = foldl(A,B,G3,B2,K2) ) ) ) ) ).

% foldl_cong
tff(fact_6548_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_6549_listrel_ONil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)),listrel(A,B,R2)) ).

% listrel.Nil
tff(fact_6550_listrel__Nil1,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xs),listrel(A,B,R2))
     => ( Xs = nil(B) ) ) ).

% listrel_Nil1
tff(fact_6551_listrel__Nil2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),nil(B)),listrel(A,B,R2))
     => ( Xs = nil(A) ) ) ).

% listrel_Nil2
tff(fact_6552_listrel__rtrancl__trans,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Zs: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel(A,A,transitive_rtrancl(A,R2)))
     => ( 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)))
       => 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_6553_listrel__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S)
     => aa(set(product_prod(list(A),list(B))),$o,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),$o),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R2)),listrel(A,B,S)) ) ).

% listrel_mono
tff(fact_6554_listrel__reflcl__if__listrel1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
     => 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_6555_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_6556_rtrancl__listrel1__if__listrel,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel(A,A,R2))
     => 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_6557_listrel__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R2)),transitive_rtrancl(list(A),listrel1(A,R2))) ).

% listrel_subset_rtrancl_listrel1
tff(fact_6558_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
           => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N)),aa(nat,B,nth(B,Ys),N)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_6559_listrel__def,axiom,
    ! [B: $tType,A: $tType,X: set(product_prod(A,B))] : listrel(A,B,X) = aa(fun(product_prod(list(A),list(B)),$o),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),$o)),fun(product_prod(list(A),list(B)),$o),product_case_prod(list(A),list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),X)))) ).

% listrel_def
tff(fact_6560_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I2: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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),I2)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),I2)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_6561_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),Xb) = none(B) )
      <=> ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% map_of_zip_is_None
tff(fact_6562_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_6563_finite__dom__map__of,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B))] : aa(set(A),$o,finite_finite2(A),dom(A,B,map_of(A,B,L))) ).

% finite_dom_map_of
tff(fact_6564_listrelp_ONil,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] : aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),nil(A)),nil(B)) ).

% listrelp.Nil
tff(fact_6565_finite__graph__map__of,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] : aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),graph(A,B,map_of(A,B,Al))) ).

% finite_graph_map_of
tff(fact_6566_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K2: B,Yb: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),Yb) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),Yb),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)) ) ).

% map_of_SomeD
tff(fact_6567_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K2: A,Xb: B,L: list(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),Xb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L))
     => ? [X3: B] : aa(A,option(B),map_of(A,B,L),K2) = aa(B,option(B),some(B),X3) ) ).

% weak_map_of_SomeI
tff(fact_6568_map__of__eq__dom,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( map_of(A,B,Xs) = map_of(A,B,Ys) )
     => ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Ys)) ) ) ).

% map_of_eq_dom
tff(fact_6569_finite__range__map__of,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A))] : aa(set(option(A)),$o,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_6570_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
      <=> ? [Y4: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),Xb) = aa(B,option(B),some(B),Y4) ) ) ).

% map_of_zip_is_Some
tff(fact_6571_map__of__eq__None__iff,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),Xb: B] :
      ( ( aa(B,option(A),map_of(B,A,Xys),Xb) = none(A) )
    <=> ~ member(B,Xb,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_6572_dom__map__of__conv__image__fst,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] : dom(A,B,map_of(A,B,Xys)) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ).

% dom_map_of_conv_image_fst
tff(fact_6573_map__of__zip__upd,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A),Xb: B,Yb: A,Z: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xs) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xs) )
       => ( ~ member(B,Xb,aa(list(B),set(B),set2(B),Xs))
         => ( ( fun_upd(B,option(A),map_of(B,A,zip(B,A,Xs,Ys)),Xb,aa(A,option(A),some(A),Yb)) = fun_upd(B,option(A),map_of(B,A,zip(B,A,Xs,Zs)),Xb,aa(A,option(A),some(A),Z)) )
           => ( map_of(B,A,zip(B,A,Xs,Ys)) = map_of(B,A,zip(B,A,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_6574_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_6575_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X: list(A),Xa: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X),Xa)
    <=> member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X),Xa),listrel(A,B,R2)) ) ).

% listrelp_listrel_eq
tff(fact_6576_ran__distinct,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
     => ( ran(A,B,map_of(A,B,Al)) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al)) ) ) ).

% ran_distinct
tff(fact_6577_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),Xb: A,Yb: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))
       => ( aa(A,option(B),map_of(A,B,Xys),Xb) = aa(B,option(B),some(B),Yb) ) ) ) ).

% map_of_is_SomeI
tff(fact_6578_map__ident,axiom,
    ! [A: $tType,X: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_adm(A,A)),X) = X ).

% map_ident
tff(fact_6579_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = nil(A) )
    <=> ( Xs = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_6580_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),map(B,A,F3),Xs) )
    <=> ( Xs = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_6581_list_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),A2) = nil(A) )
    <=> ( A2 = nil(B) ) ) ).

% list.map_disc_iff
tff(fact_6582_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),Xs: list(C)] : aa(list(B),list(A),map(B,A,F3),aa(list(C),list(B),map(C,B,G3),Xs)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G3)),Xs) ).

% map_map
tff(fact_6583_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),List: list(C)] : aa(list(B),list(A),map(B,A,F3),aa(list(C),list(B),map(C,B,G3),List)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G3)),List) ).

% List.map.compositionality
tff(fact_6584_list_Omap__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: fun(B,A),F3: fun(C,B),V: list(C)] : aa(list(B),list(A),map(B,A,G3),aa(list(C),list(B),map(C,B,F3),V)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,G3),F3)),V) ).

% list.map_comp
tff(fact_6585_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),G3: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,G3),Xs) )
    <=> ! [X4: B] :
          ( member(B,X4,aa(list(B),set(B),set2(B),Xs))
         => ( aa(B,A,F3,X4) = aa(B,A,G3,X4) ) ) ) ).

% map_eq_conv
tff(fact_6586_length__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_6587_list_Omap__id0,axiom,
    ! [A: $tType] : map(A,A,id(A)) = id(list(A)) ).

% list.map_id0
tff(fact_6588_map__replicate,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Nb: nat,Xb: B] : aa(list(B),list(A),map(B,A,F3),replicate(B,Nb,Xb)) = replicate(A,Nb,aa(B,A,F3,Xb)) ).

% map_replicate
tff(fact_6589_list_Oset__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),V: list(B)] : aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F3),V)) = aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),V)) ).

% list.set_map
tff(fact_6590_map__snd__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(A),map(product_prod(nat,A),A,product_snd(nat,A)),enumerate(A,Nb,Xs)) = Xs ).

% map_snd_enumerate
tff(fact_6591_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,F3),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_map_eq_map
tff(fact_6592_map__fun__upd,axiom,
    ! [B: $tType,A: $tType,Yb: A,Xs: list(A),F3: fun(A,B),V: B] :
      ( ~ member(A,Yb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(list(A),list(B),map(A,B,fun_upd(A,B,F3,Yb,V)),Xs) = aa(list(A),list(B),map(A,B,F3),Xs) ) ) ).

% map_fun_upd
tff(fact_6593_List_Omap_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),G3: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),comp(list(C),list(B),list(A),map(C,B,F3)),map(A,C,G3)) = map(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G3)) ).

% List.map.comp
tff(fact_6594_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),G3: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),comp(list(C),list(B),list(A),map(C,B,F3)),map(A,C,G3)) = map(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G3)) ).

% map_comp_map
tff(fact_6595_size__list__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F3),aa(list(B),list(A),map(B,A,G3),Xs)) = aa(list(B),nat,size_list(B,aa(fun(B,A),fun(B,nat),comp(A,nat,B,F3),G3)),Xs) ).

% size_list_map
tff(fact_6596_nth__map,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F3: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F3),Xs)),Nb) = aa(A,B,F3,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% nth_map
tff(fact_6597_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),map(product_prod(A,B),A,product_fst(A,B)),zip(A,B,Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_6598_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),map(product_prod(A,B),B,product_snd(A,B)),zip(A,B,Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_6599_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_6600_inj__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A))))
    <=> inj_on(A,B,F3,top_top(set(A))) ) ).

% inj_map
tff(fact_6601_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Xb: A,Yb: B] :
      ( distinct(A,aa(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),Xb) = aa(B,option(B),some(B),Yb) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).

% map_of_eq_Some_iff
tff(fact_6602_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Yb: B,Xb: A] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(B,option(B),some(B),Yb) = aa(A,option(B),map_of(A,B,Xys),Xb) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).

% Some_eq_map_of_iff
tff(fact_6603_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G3: fun(A,fun(B,A)),A2: A,F3: fun(C,B),Xs: list(C)] : foldl(A,B,G3,A2,aa(list(C),list(B),map(C,B,F3),Xs)) = foldl(A,C,aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_adn(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G3),F3),A2,Xs) ).

% foldl_map
tff(fact_6604_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,nat),G3: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),nat),comp(list(B),nat,list(A),size_list(B,F3)),map(A,B,G3)) = size_list(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F3),G3)) ).

% list.size_gen_o_map
tff(fact_6605_nths__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),I6: set(nat)] : nths(A,aa(list(B),list(A),map(B,A,F3),Xs),I6) = aa(list(B),list(A),map(B,A,F3),nths(B,Xs,I6)) ).

% nths_map
tff(fact_6606_list_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] : aa(list(B),list(A),map(B,A,F3),nil(B)) = nil(A) ).

% list.simps(8)
tff(fact_6607_map__concat,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(list(B))] : aa(list(B),list(A),map(B,A,F3),concat(B,Xs)) = concat(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),Xs)) ).

% map_concat
tff(fact_6608_list_Omap__ident,axiom,
    ! [A: $tType,Ta: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_adm(A,A)),Ta) = Ta ).

% list.map_ident
tff(fact_6609_rotate1__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate1(A),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rotate1(B),Xs)) ).

% rotate1_map
tff(fact_6610_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,A),Xs: list(B),G3: fun(C,A),Ys: list(C)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(C),list(A),map(C,A,G3),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_6611_remdups__map__remdups,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : remdups(A,aa(list(B),list(A),map(B,A,F3),remdups(B,Xs))) = remdups(A,aa(list(B),list(A),map(B,A,F3),Xs)) ).

% remdups_map_remdups
tff(fact_6612_list_Omap__id,axiom,
    ! [A: $tType,Ta: list(A)] : aa(list(A),list(A),map(A,A,id(A)),Ta) = Ta ).

% list.map_id
tff(fact_6613_List_Omap_Oidentity,axiom,
    ! [A: $tType] : map(A,A,aTP_Lamp_adm(A,A)) = id(list(A)) ).

% List.map.identity
tff(fact_6614_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),map(product_prod(A,B),A,product_fst(A,B)),Xs) = aa(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),map(product_prod(A,B),B,product_snd(A,B)),Xs) = aa(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_6615_enumerate__Suc__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : enumerate(A,aa(nat,nat,suc,Nb),Xs) = aa(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,Nb,Xs)) ).

% enumerate_Suc_eq
tff(fact_6616_map__injective,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ( inj_on(B,A,F3,top_top(set(B)))
       => ( Xs = Ys ) ) ) ).

% map_injective
tff(fact_6617_take__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F3: fun(B,A),Xs: list(B)] : take(A,Nb,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),take(B,Nb,Xs)) ).

% take_map
tff(fact_6618_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Yaa: list(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( Xb = Yaa )
     => ( ! [Z2: A] :
            ( member(A,Z2,aa(list(A),set(A),set2(A),Yaa))
           => ( aa(A,B,F3,Z2) = aa(A,B,G3,Z2) ) )
       => ( aa(list(A),list(B),map(A,B,F3),Xb) = aa(list(A),list(B),map(A,B,G3),Yaa) ) ) ) ).

% list.map_cong
tff(fact_6619_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ! [Z2: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
         => ( aa(A,B,F3,Z2) = aa(A,B,G3,Z2) ) )
     => ( aa(list(A),list(B),map(A,B,F3),Xb) = aa(list(A),list(B),map(A,B,G3),Xb) ) ) ).

% list.map_cong0
tff(fact_6620_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Xaa: list(A),F3: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z2: A,Za2: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
         => ( member(A,Za2,aa(list(A),set(A),set2(A),Xaa))
           => ( ( aa(A,B,F3,Z2) = aa(A,B,Fa,Za2) )
             => ( Z2 = Za2 ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F3),Xb) = aa(list(A),list(B),map(A,B,Fa),Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_6621_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,B,F3,X3) = aa(A,B,G3,X3) ) )
     => ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,G3),Xs) ) ) ).

% map_ext
tff(fact_6622_map__idI,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,A,F3,X3) = X3 ) )
     => ( aa(list(A),list(A),map(A,A,F3),Xs) = Xs ) ) ).

% map_idI
tff(fact_6623_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,B,F3,X3) = aa(A,B,G3,X3) ) )
       => ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,G3),Ys) ) ) ) ).

% map_cong
tff(fact_6624_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys: list(B),F3: fun(A,B)] :
      ( ? [Xs3: list(A)] : Ys = aa(list(A),list(B),map(A,B,F3),Xs3)
    <=> ! [X4: B] :
          ( member(B,X4,aa(list(B),set(B),set2(B),Ys))
         => ? [Xa2: A] : X4 = aa(A,B,F3,Xa2) ) ) ).

% ex_map_conv
tff(fact_6625_map__update,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),K2: nat,Yb: B] : aa(list(B),list(A),map(B,A,F3),list_update(B,Xs,K2,Yb)) = list_update(A,aa(list(B),list(A),map(B,A,F3),Xs),K2,aa(B,A,F3,Yb)) ).

% map_update
tff(fact_6626_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K2: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_kj(A,fun(B,A)),K2)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K2) ).

% map_replicate_const
tff(fact_6627_image__set,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F3),Xs)) ).

% image_set
tff(fact_6628_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F3: fun(D,B),Xs: list(D),G3: fun(D,C)] : aa(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),map(D,B,F3),Xs),aa(list(D),list(C),map(D,C,G3),Xs))) = aa(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_ado(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F3),G3)),Xs) ).

% map2_map_map
tff(fact_6629_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(product_prod(B,C),A),Xs: list(B),G3: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G3),Ys))) = aa(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_adp(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F3),G3))),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_6630_zip__map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,F3: fun(C,A),Xs: list(C),G3: fun(D,B),Ys: list(D)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),aa(list(D),list(B),map(D,B,G3),Ys)) = aa(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_adq(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3))),zip(C,D,Xs,Ys)) ).

% zip_map_map
tff(fact_6631_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(product_prod(B,C),A),G3: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,aa(list(D),list(B),map(D,B,G3),Xs),Ys)) = aa(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_adr(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F3),G3))),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_6632_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F3: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F3),Ys)) = aa(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_ads(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_6633_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),Ys) = aa(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_adt(fun(C,A),fun(C,fun(B,product_prod(A,B))),F3))),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_6634_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ( inj_on(B,A,F3,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_6635_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F3,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),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,F3),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_on_map_eq_map
tff(fact_6636_distinct__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
    <=> ( distinct(B,Xs)
        & inj_on(B,A,F3,aa(list(B),set(B),set2(B),Xs)) ) ) ).

% distinct_map
tff(fact_6637_remdups__adj__map__injective,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( remdups_adj(B,aa(list(A),list(B),map(A,B,F3),Xs)) = aa(list(A),list(B),map(A,B,F3),remdups_adj(A,Xs)) ) ) ).

% remdups_adj_map_injective
tff(fact_6638_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),map(product_prod(A,B),A,product_fst(A,B)),Zs),aa(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_6639_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),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),map(product_prod(A,B),A,product_fst(A,B)),Ys)) )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),aa(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),X3) = aa(A,option(B),map_of(A,B,Ys),X3) ) )
       => ( map_of(A,B,Xs) = map_of(A,B,Ys) ) ) ) ).

% map_of_eqI
tff(fact_6640_map__removeAll__inj,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xb: A,Xs: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),removeAll(A,Xb),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F3,Xb)),aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ).

% map_removeAll_inj
tff(fact_6641_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A))))
     => inj_on(A,B,F3,top_top(set(A))) ) ).

% inj_mapD
tff(fact_6642_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))),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_adv(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_6643_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))),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_adw(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_6644_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)),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_ym(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_6645_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),map(product_prod(A,B),A,product_fst(A,B)),Zs) = Xs )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs) = Ys ) ) ) ) ).

% zip_eq_conv
tff(fact_6646_distinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xb: B,Xs: list(B)] :
          ( distinct(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),Xs)))
        <=> ( ~ member(A,aa(B,A,F3,Xb),aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),Xs)))
            & distinct(A,aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ) ).

% distinct_insort_key
tff(fact_6647_map__removeAll__inj__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xb: A,Xs: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),removeAll(A,Xb),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F3,Xb)),aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ).

% map_removeAll_inj_on
tff(fact_6648_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K2: A,V1: B,V22: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V1),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),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_6649_map__of__inject__set,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys))
       => ( ( map_of(A,B,Xs) = map_of(A,B,Ys) )
        <=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Ys) ) ) ) ) ).

% map_of_inject_set
tff(fact_6650_inj__on__mapI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(list(A))] :
      ( inj_on(A,B,F3,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)),A5)))
     => inj_on(list(A),list(B),map(A,B,F3),A5) ) ).

% inj_on_mapI
tff(fact_6651_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F3: fun(A,B),X: A] :
      aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F3),Xs))),X) = $ite(member(A,X,aa(list(A),set(A),set2(A),Xs)),aa(B,option(B),some(B),aa(A,B,F3,X)),none(B)) ).

% map_of_zip_map
tff(fact_6652_map__fst__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),zip(A,B,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)),Xs) ).

% map_fst_zip_take
tff(fact_6653_map__snd__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(B),Ys: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),zip(B,A,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)),Ys) ).

% map_snd_zip_take
tff(fact_6654_graph__map__of__if__distinct__dom,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
     => ( graph(A,B,map_of(A,B,Al)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al) ) ) ).

% graph_map_of_if_distinct_dom
tff(fact_6655_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),Ta: list(product_prod(A,C)),K2: A,Xb: C] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,Ta),K2) = aa(C,option(C),some(C),Xb) )
       => ( aa(B,option(C),map_of(B,C,aa(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_adx(fun(A,B),fun(A,fun(C,product_prod(B,C))),F3))),Ta)),aa(A,B,F3,K2)) = aa(C,option(C),some(C),Xb) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_6656_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_ady(list(product_prod(A,B)),fun(A,fun(B,$o)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_6657_set__relcomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_aea(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_6658_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_ts(A,fun(A,$o))) = ord_min(A) ) ) ).

% dual_max
tff(fact_6659_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y22: 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),Y22) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
tff(fact_6660_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)),insert(A),X21),aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_6661_nth__Cons__Suc,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% nth_Cons_Suc
tff(fact_6662_nth__Cons__0,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),zero_zero(nat)) = Xb ).

% nth_Cons_0
tff(fact_6663_take__Suc__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : take(A,aa(nat,nat,suc,Nb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),take(A,Nb,Xs)) ).

% take_Suc_Cons
tff(fact_6664_nths__singleton,axiom,
    ! [A: $tType,Xb: A,A5: set(nat)] :
      nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)),A5) = $ite(member(nat,zero_zero(nat),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)),nil(A)) ).

% nths_singleton
tff(fact_6665_Cons__listrel1__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)),listrel1(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
          & ( Xs = Ys ) )
        | ( ( Xb = Yb )
          & 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_6666_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xb: B,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F3,Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),groups4207007520872428315er_sum(B,A,F3,A2,Xs))) ) ).

% horner_sum_simps(2)
tff(fact_6667_lexord__cons__cons,axiom,
    ! [A: $tType,A2: A,Xb: list(A),B2: A,Yb: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Xb)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Yb)),lexord(A,R2))
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
        | ( ( A2 = B2 )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Yb),lexord(A,R2)) ) ) ) ).

% lexord_cons_cons
tff(fact_6668_lexord__Nil__left,axiom,
    ! [A: $tType,Yb: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Yb),lexord(A,R2))
    <=> ? [A7: A,X4: list(A)] : Yb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),X4) ) ).

% lexord_Nil_left
tff(fact_6669_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : enumerate(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(product_prod(nat,A),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),cons(product_prod(nat,A)),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Nb),Xb)),enumerate(A,aa(nat,nat,suc,Nb),Xs)) ).

% enumerate_simps(2)
tff(fact_6670_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),A2: A,As: list(A),B2: B,Bs: list(B)] : map_upds(A,B,Mb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),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),Mb,A2,aa(B,option(B),some(B),B2)),As,Bs) ).

% map_upds_Cons
tff(fact_6671_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Yb: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Yb),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb)),zip(A,B,Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_6672_nth__Cons__numeral,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_6673_take__Cons__numeral,axiom,
    ! [A: $tType,V: num,Xb: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_6674_Cons__in__lex,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)),lex(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( Xb = Yb )
          & 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_6675_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_aeb(fun(B,A),fun(B,list(A)),F3)),Xs)) = aa(list(B),list(A),map(B,A,F3),Xs) ).

% concat_map_singleton
tff(fact_6676_nth__Cons__pos,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6677_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: 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)) = concat(list(A),aa(list(A),list(list(list(A))),map(A,list(list(A)),aTP_Lamp_aec(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_6678_list_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X21: B,X222: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X21),X222)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F3,X21)),aa(list(B),list(A),map(B,A,F3),X222)) ).

% list.simps(9)
tff(fact_6679_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),F3: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ? [Z2: B,Zs2: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z2),Zs2) )
          & ( Xb = aa(B,A,F3,Z2) )
          & ( Xs = aa(list(B),list(A),map(B,A,F3),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_6680_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xs: list(B),Yb: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys) )
     => ? [Z2: B,Zs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z2),Zs2) )
          & ( aa(B,A,F3,Z2) = Yb )
          & ( aa(list(B),list(A),map(B,A,F3),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_6681_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),F3: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & ( Xb = aa(B,A,F3,Z5) )
          & ( Xs = aa(list(B),list(A),map(B,A,F3),Zs3) ) ) ) ).

% Cons_eq_map_conv
tff(fact_6682_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xs: list(B),Yb: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys) )
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & ( aa(B,A,F3,Z5) = Yb )
          & ( aa(list(B),list(A),map(B,A,F3),Zs3) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_6683_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,Nb),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_aee(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,Nb,Xs))) ).

% n_lists.simps(2)
tff(fact_6684_List_Obind__def,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F3: fun(B,list(A))] : bind(B,A,Xs,F3) = concat(A,aa(list(B),list(list(A)),map(B,list(A),F3),Xs)) ).

% List.bind_def
tff(fact_6685_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_aef(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_6686_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_yk(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_6687_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) )
     => ~ ! [X3: A,Xs4: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
           => ! [Y: B,Ys5: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys5) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) )
                 => ( Xys != zip(A,B,Xs4,Ys5) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6688_inj__split__Cons,axiom,
    ! [A: $tType,X5: 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_aed(list(A),fun(A,list(A)))),X5) ).

% inj_split_Cons
tff(fact_6689_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs))) ).

% set_subset_Cons
tff(fact_6690_remdups_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),remdups(A,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),remdups(A,Xs))) ).

% remdups.simps(2)
tff(fact_6691_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs))
    <=> ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
        & distinct(A,Xs) ) ) ).

% distinct.simps(2)
tff(fact_6692_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I2: nat,V: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),I2,V) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_aeg(A,fun(list(A),fun(A,fun(nat,list(A)))),Xb),Xs),V),I2) ).

% list_update.simps(2)
tff(fact_6693_set__ConsD,axiom,
    ! [A: $tType,Yb: A,Xb: A,Xs: list(A)] :
      ( member(A,Yb,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)))
     => ( ( Yb = Xb )
        | member(A,Yb,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_ConsD
tff(fact_6694_list_Oset__cases,axiom,
    ! [A: $tType,E3: A,A2: list(A)] :
      ( member(A,E3,aa(list(A),set(A),set2(A),A2))
     => ( ! [Z23: list(A)] : A2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z12),Z23) )
             => ~ member(A,E3,aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_6695_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : 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_6696_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Yb: A,X222: list(A),X21: A] :
      ( member(A,Yb,aa(list(A),set(A),set2(A),X222))
     => member(A,Yb,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_6697_replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : replicate(A,aa(nat,nat,suc,Nb),Xb) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),replicate(A,Nb,Xb)) ).

% replicate_Suc
tff(fact_6698_list__update__code_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),zero_zero(nat),Yb) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Xs) ).

% list_update_code(2)
tff(fact_6699_list__update__code_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I2: nat,Yb: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(nat,nat,suc,I2),Yb) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),list_update(A,Xs,I2,Yb)) ).

% list_update_code(3)
tff(fact_6700_arg__min__list_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A] : arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) = Xb ) ).

% arg_min_list.simps(1)
tff(fact_6701_listrel1I2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Xb: A] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys)),listrel1(A,R2)) ) ).

% listrel1I2
tff(fact_6702_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,B),Bs2: list(B)] : Xb != 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))),F2),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))
     => ~ ! [F2: fun(A,B),A4: A,As2: list(A),Bs2: list(B)] : Xb != 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))),F2),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_6703_successively_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P6: fun(A,fun(A,$o))] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P6),nil(A))
     => ( ! [P6: fun(A,fun(A,$o)),X3: A] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ~ ! [P6: fun(A,fun(A,$o)),X3: A,Y: A,Xs2: list(A)] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2))) ) ) ).

% successively.cases
tff(fact_6704_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: product_prod(fun(A,B),list(A))] :
          ( ! [F2: fun(A,B),X3: A] : Xb != 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)),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
         => ( ! [F2: fun(A,B),X3: A,Y: A,Zs2: list(A)] : Xb != 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)),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2)))
           => ~ ! [A4: fun(A,B)] : Xb != 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_6705_sorted__wrt_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P6: fun(A,fun(A,$o))] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P6),nil(A))
     => ~ ! [P6: fun(A,fun(A,$o)),X3: A,Ys3: list(A)] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3)) ) ).

% sorted_wrt.cases
tff(fact_6706_shuffles_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : Xb != 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)] : Xb != 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))
       => ~ ! [X3: A,Xs2: list(A),Y: A,Ys3: list(A)] : Xb != 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),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)) ) ) ).

% shuffles.cases
tff(fact_6707_splice_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : Xb != 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)
     => ~ ! [X3: A,Xs2: list(A),Ys3: list(A)] : Xb != 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),X3),Xs2)),Ys3) ) ).

% splice.cases
tff(fact_6708_remdups__adj_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Yb: list(A)] :
      ( ( remdups_adj(A,Xb) = Yb )
     => ( ( ( Xb = nil(A) )
         => ( Yb != nil(A) ) )
       => ( ! [X3: A] :
              ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
             => ( Yb != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) ) )
         => ~ ! [X3: A,Y: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2)) )
               => ( Yb != $ite(X3 = Y,remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2)))) ) ) ) ) ) ).

% remdups_adj.elims
tff(fact_6709_remdups__adj_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) ).

% remdups_adj.simps(2)
tff(fact_6710_shufflesE,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => ( ( ( Zs = Xs )
         => ( Ys != nil(A) ) )
       => ( ( ( Zs = Ys )
           => ( Xs != nil(A) ) )
         => ( ! [X3: A,Xs4: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
               => ! [Z2: A,Zs4: list(A)] :
                    ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs4) )
                   => ( ( X3 = Z2 )
                     => ~ member(list(A),Zs4,shuffles(A,Xs4,Ys)) ) ) )
           => ~ ! [Y: A,Ys5: list(A)] :
                  ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys5) )
                 => ! [Z2: A,Zs4: list(A)] :
                      ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs4) )
                     => ( ( Y = Z2 )
                       => ~ member(list(A),Zs4,shuffles(A,Xs,Ys5)) ) ) ) ) ) ) ) ).

% shufflesE
tff(fact_6711_insort__key_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) ) ).

% insort_key.simps(1)
tff(fact_6712_distinct__singleton,axiom,
    ! [A: $tType,Xb: A] : distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) ).

% distinct_singleton
tff(fact_6713_inj__on__Cons1,axiom,
    ! [A: $tType,Xb: A,A5: set(list(A))] : inj_on(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),A5) ).

% inj_on_Cons1
tff(fact_6714_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [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) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_6715_Suc__length__conv,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,Nb) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [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) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_6716_Cons__in__shuffles__leftI,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z: A] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => 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_6717_Cons__in__shuffles__rightI,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z: A] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => 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_6718_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_6719_not__Cons__self2,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) != Xs ).

% not_Cons_self2
tff(fact_6720_ord_Omax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : max(A,Less_eq) = max(A,Less_eq) ).

% ord.max.cong
tff(fact_6721_ord_Omax__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),A2: A,B2: A] :
      aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),Less_eq,A2),B2),B2,A2) ).

% ord.max_def
tff(fact_6722_remove1_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Yb: A,Xs: list(A)] :
      remove1(A,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Xs)) = $ite(Xb = Yb,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),remove1(A,Xb,Xs))) ).

% remove1.simps(2)
tff(fact_6723_removeAll_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Yb: A,Xs: list(A)] :
      aa(list(A),list(A),removeAll(A,Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Xs)) = $ite(Xb = Yb,aa(list(A),list(A),removeAll(A,Xb),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),aa(list(A),list(A),removeAll(A,Xb),Xs))) ).

% removeAll.simps(2)
tff(fact_6724_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_6725_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_6726_list_Oexhaust,axiom,
    ! [A: $tType,Yb: list(A)] :
      ( ( Yb != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : Yb != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X212),X223) ) ).

% list.exhaust
tff(fact_6727_min__list_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A)] :
          ( ! [X3: A,Xs2: list(A)] : Xb != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)
         => ( Xb = nil(A) ) ) ) ).

% min_list.cases
tff(fact_6728_transpose_Ocases,axiom,
    ! [A: $tType,Xb: list(list(A))] :
      ( ( Xb != nil(list(A)) )
     => ( ! [Xss2: list(list(A))] : Xb != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)
       => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] : Xb != 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),X3),Xs2)),Xss2) ) ) ).

% transpose.cases
tff(fact_6729_remdups__adj_Ocases,axiom,
    ! [A: $tType,Xb: list(A)] :
      ( ( Xb != nil(A) )
     => ( ! [X3: A] : Xb != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))
       => ~ ! [X3: A,Y: A,Xs2: list(A)] : Xb != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_6730_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_6731_list__induct2_H,axiom,
    ! [A: $tType,B: $tType,P2: fun(list(A),fun(list(B),$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,nil(A)),nil(B))
     => ( ! [X3: A,Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),nil(B))
       => ( ! [Y: B,Ys3: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3))
         => ( ! [X3: A,Xs2: list(A),Y: B,Ys3: list(B)] :
                ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs2),Ys3)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3)) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_6732_list__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(list(A),$o)] :
      ( ( Xs != nil(A) )
     => ( ! [X3: A] : aa(list(A),$o,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ( ! [X3: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P2,Xs2)
               => aa(list(A),$o,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)) ) )
         => aa(list(A),$o,P2,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_6733_remdups__adj_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Yb: A,Xs: list(A)] :
      remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Xs))) = $ite(Xb = Yb,remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Xs)))) ).

% remdups_adj.simps(3)
tff(fact_6734_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P2: fun(list(A),fun(list(B),$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,nil(A)),nil(B))
       => ( ! [X3: A,Xs2: list(A),Y: B,Ys3: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs2),Ys3)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_6735_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P2: fun(list(A),fun(list(B),fun(list(C),$o)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P2,nil(A)),nil(B)),nil(C))
         => ( ! [X3: A,Xs2: list(A),Y: B,Ys3: list(B),Z2: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P2,Xs2),Ys3),Zs2)
                   => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z2),Zs2)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P2,Xs),Ys),Zs) ) ) ) ) ).

% list_induct3
tff(fact_6736_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws2: list(D),P2: fun(list(A),fun(list(B),fun(list(C),fun(list(D),$o))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D),nat,size_size(list(D)),Ws2) )
         => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P2,nil(A)),nil(B)),nil(C)),nil(D))
           => ( ! [X3: A,Xs2: list(A),Y: B,Ys3: list(B),Z2: 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) )
                     => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P2,Xs2),Ys3),Zs2),Ws)
                       => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z2),Zs2)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),W),Ws)) ) ) ) )
             => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P2,Xs),Ys),Zs),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_6737_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: A,Ys: list(A)] : shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),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),Xb)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys))) ).

% shuffles.simps(3)
tff(fact_6738_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Yb: A,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb)),shuffles(A,Xs,Ys))),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys))) ).

% Cons_shuffles_subset2
tff(fact_6739_Cons__shuffles__subset1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb)),shuffles(A,Xs,Ys))),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys)) ).

% Cons_shuffles_subset1
tff(fact_6740_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A,Yb: A,Ys: list(A)] :
          aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,F3,Yb)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Ys))) ) ).

% insort_key.simps(2)
tff(fact_6741_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))
     => ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys) ) ) ).

% impossible_Cons
tff(fact_6742_listrelp_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: list(A),A22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),A1),A22)
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X4: A,Y4: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys4) )
            & aa(B,$o,aa(A,fun(B,$o),R2,X4),Y4)
            & aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs3),Ys4) ) ) ) ).

% listrelp.simps
tff(fact_6743_listrelp_Ocases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: list(A),A22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),A1),A22)
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X3: A,Y: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3) )
                 => ( aa(B,$o,aa(A,fun(B,$o),R2,X3),Y)
                   => ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs2),Ys3) ) ) ) ) ) ).

% listrelp.cases
tff(fact_6744_listrelp_OCons,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xb: A,Yb: B,Xs: list(A),Ys: list(B)] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,Xb),Yb)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs),Ys)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Yb),Ys)) ) ) ).

% listrelp.Cons
tff(fact_6745_foldl__Cons,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,A)),A2: A,Xb: B,Xs: list(B)] : foldl(A,B,F3,A2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)) = foldl(A,B,F3,aa(B,A,aa(A,fun(B,A),F3,A2),Xb),Xs) ).

% foldl_Cons
tff(fact_6746_take__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_aeh(A,fun(list(A),fun(nat,list(A))),Xb),Xs),Nb) ).

% take_Cons
tff(fact_6747_Cons__in__subseqsD,axiom,
    ! [A: $tType,Yb: A,Ys: list(A),Xs: list(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))
     => member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) ) ).

% Cons_in_subseqsD
tff(fact_6748_nth__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = case_nat(A,Xb,nth(A,Xs),Nb) ).

% nth_Cons
tff(fact_6749_ord_Olexordp_Omono,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_aei(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Less)) ).

% ord.lexordp.mono
tff(fact_6750_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A,Yb: A,Zs: list(A)] :
          arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Zs))) = $let(
            m2: A,
            m2:= arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Zs)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,F3,m2)),Xb,m2) ) ) ).

% arg_min_list.simps(2)
tff(fact_6751_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X4: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_6752_distinct__set__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(set(A),aa(list(list(A)),list(set(A)),map(list(A),set(A),set2(A)),subseqs(A,Xs))) ) ).

% distinct_set_subseqs
tff(fact_6753_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B),A2: A] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),aa(A,B,F3,X3)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),A2),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_6754_map__of__Cons__code_I2_J,axiom,
    ! [A: $tType,B: $tType,L: B,V: A,Ps: list(product_prod(B,A)),K2: B] :
      aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),L),V)),Ps)),K2) = $ite(L = K2,aa(A,option(A),some(A),V),aa(B,option(A),map_of(B,A,Ps),K2)) ).

% map_of_Cons_code(2)
tff(fact_6755_listrel1I1,axiom,
    ! [A: $tType,Xb: A,Yb: A,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Xs)),listrel1(A,R2)) ) ).

% listrel1I1
tff(fact_6756_Cons__listrel1E1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Ys),listrel1(A,R2))
     => ( ! [Y: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs) )
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Zs2) )
             => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_6757_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Yb: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)),listrel1(A,R2))
     => ( ! [X3: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys) )
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Yb),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Zs2) )
             => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs2),Ys),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_6758_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,Xb: A,Yb: B,R2: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb),R2)
     => ( 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))
       => member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Yb),Ys)),listrel(A,B,R2)) ) ) ).

% listrel.Cons
tff(fact_6759_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Yb: A,Ys: list(A),Xs: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)),Xs),listrel(A,B,R2))
     => ~ ! [Y: B,Ys3: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3) )
           => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Yb),Y),R2)
             => ~ 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_6760_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Yb: B,Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Yb),Ys)),listrel(A,B,R2))
     => ~ ! [X3: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
           => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Yb),R2)
             => ~ 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_6761_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: A] :
      aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Yb) = $ite(Xb = Yb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Yb)),one_one(nat)),aa(A,nat,count_list(A,Xs),Yb)) ).

% count_list.simps(2)
tff(fact_6762_the__elem__set,axiom,
    ! [A: $tType,Xb: 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),Xb),nil(A)))) = Xb ).

% the_elem_set
tff(fact_6763_rtrancl__listrel1__ConsI1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Xb: A] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R2)))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ).

% rtrancl_listrel1_ConsI1
tff(fact_6764_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_aej(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_6765_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xb: A,Ys: list(B)] : zip(A,B,replicate(A,Nb,Xb),Ys) = aa(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),Xb)),take(B,Nb,Ys)) ).

% zip_replicate1
tff(fact_6766_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_6767_nth__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ).

% nth_Cons'
tff(fact_6768_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Nb: nat,Yb: B] : zip(A,B,Xs,replicate(B,Nb,Yb)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ym(B,fun(A,product_prod(A,B))),Yb)),take(A,Nb,Xs)) ).

% zip_replicate2
tff(fact_6769_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X4: A,Y4: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys4) )
            & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4),R2)
            & 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_6770_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X3: A,Y: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3) )
                 => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),R2)
                   => ~ 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_6771_remdups__adj__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      remdups_adj(A,replicate(A,Nb,Xb)) = $ite(Nb = zero_zero(nat),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) ).

% remdups_adj_replicate
tff(fact_6772_remdups__adj__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ( remdups_adj(A,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) )
     => ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) ) ) ).

% remdups_adj_singleton
tff(fact_6773_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)),map(A,product_prod(A,A),aTP_Lamp_yk(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_6774_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X21: A,X222: list(A)] : aa(list(A),nat,size_list(A,Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X21)),aa(list(A),nat,size_list(A,Xb),X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6775_shuffles_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Yb: set(list(A))] :
      ( ( shuffles(A,Xb,Xaa) = Yb )
     => ( ( ( Xb = nil(A) )
         => ( Yb != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xaa),bot_bot(set(list(A)))) ) )
       => ( ( ( Xaa = nil(A) )
           => ( Yb != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xb),bot_bot(set(list(A)))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ! [Y: A,Ys3: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3) )
                   => ( Yb != 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),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)))),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),X3),Xs2),Ys3))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_6776_nth__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6777_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Yb: A,Xs: list(A),Nb: nat] :
      ( ( Xb != Yb )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = Yb )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Yb )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6778_take__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      take(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(Nb = zero_zero(nat),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs))) ).

% take_Cons'
tff(fact_6779_Cons__replicate__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat,Yb: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = replicate(A,Nb,Yb) )
    <=> ( ( Xb = Yb )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_6780_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_za(fun(A,B),fun(A,product_prod(A,B)),F3)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F3),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_6781_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Mb: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,Mb) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_acl(fun(A,option(B)),fun(A,product_prod(A,B)),Mb)),Xs)) = Mb ) ) ).

% map_of_map_keys
tff(fact_6782_Cons__lenlex__iff,axiom,
    ! [A: $tType,Mb: A,Ms: list(A),Nb: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Mb),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Nb),Ns)),lenlex(A,R2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Mb),Nb),R2) )
        | ( ( Mb = Nb )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6783_map__of_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,P: 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)),P),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P))) ).

% map_of.simps(2)
tff(fact_6784_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,Xb: A,Yb: A,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_rtrancl(A,R2))
     => ( 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)))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_6785_Pow__set_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      pow2(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs))) = $let(
        a3: set(set(A)),
        a3:= pow2(A,aa(list(A),set(A),set2(A),Xs)),
        aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a3),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb)),a3)) ) ).

% Pow_set(2)
tff(fact_6786_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),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)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_6787_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_aef(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_6788_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A5: set(A)] : set_Cons(A,A5,aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_aek(A,list(A))),A5) ).

% set_Cons_sing_Nil
tff(fact_6789_n__lists__Nil,axiom,
    ! [A: $tType,Nb: nat] :
      n_lists(A,Nb,nil(A)) = $ite(Nb = zero_zero(nat),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))),nil(list(A))) ).

% n_lists_Nil
tff(fact_6790_listset_Osimps_I2_J,axiom,
    ! [A: $tType,A5: 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)),A5),As3)) = set_Cons(A,A5,listset(A,As3)) ).

% listset.simps(2)
tff(fact_6791_subseqs_Osimps_I1_J,axiom,
    ! [A: $tType] : 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_6792_product__lists_Osimps_I1_J,axiom,
    ! [A: $tType] : 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_6793_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_6794_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I2)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_6795_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_6796_set__Cons__def,axiom,
    ! [A: $tType,A5: set(A),XS: set(list(A))] : set_Cons(A,A5,XS) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(list(A)),fun(list(A),$o),aTP_Lamp_ael(set(A),fun(set(list(A)),fun(list(A),$o)),A5),XS)) ).

% set_Cons_def
tff(fact_6797_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xb: 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),Xb),nil(A))),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% concat_inth
tff(fact_6798_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),Mb: fun(A,option(B)),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))
     => ( map_upds(A,B,Mb,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),Xb),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,Mb,Xs,Ys),Xb,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_6799_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_6800_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_6801_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_6802_append_Oassoc,axiom,
    ! [A: $tType,A2: list(A),B2: list(A),C3: 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)),C3) = 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),C3)) ).

% append.assoc
tff(fact_6803_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_6804_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_6805_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_6806_self__append__conv,axiom,
    ! [A: $tType,Yb: list(A),Ys: list(A)] :
      ( ( Yb = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Yb),Ys) )
    <=> ( Ys = nil(A) ) ) ).

% self_append_conv
tff(fact_6807_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_6808_self__append__conv2,axiom,
    ! [A: $tType,Yb: list(A),Xs: list(A)] :
      ( ( Yb = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Yb) )
    <=> ( Xs = nil(A) ) ) ).

% self_append_conv2
tff(fact_6809_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_6810_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_6811_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_6812_map__append,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] : aa(list(B),list(A),map(B,A,F3),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),map(B,A,F3),Xs)),aa(list(B),list(A),map(B,A,F3),Ys)) ).

% map_append
tff(fact_6813_concat__append,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(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),concat(A,Xs)),concat(A,Ys)) ).

% concat_append
tff(fact_6814_removeAll__append,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),removeAll(A,Xb),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,Xb),Xs)),aa(list(A),list(A),removeAll(A,Xb),Ys)) ).

% removeAll_append
tff(fact_6815_foldl__append,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,A)),A2: A,Xs: list(B),Ys: list(B)] : foldl(A,B,F3,A2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = foldl(A,B,F3,foldl(A,B,F3,A2,Xs),Ys) ).

% foldl_append
tff(fact_6816_append1__eq__conv,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Yb: 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),Xb),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),Yb),nil(A))) )
    <=> ( ( Xs = Ys )
        & ( Xb = Yb ) ) ) ).

% append1_eq_conv
tff(fact_6817_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_6818_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_6819_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_6820_size__list__append,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F3),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,F3),Xs)),aa(list(A),nat,size_list(A,F3),Ys)) ).

% size_list_append
tff(fact_6821_bind__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xb: B,Xs: list(B),F3: fun(B,list(A))] : bind(B,A,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs),F3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F3,Xb)),bind(B,A,Xs,F3)) ).

% bind_simps(2)
tff(fact_6822_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: 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),Xb),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% nth_append_length
tff(fact_6823_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: 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)),Nb)) = aa(nat,A,nth(A,Ys),Nb) ).

% nth_append_length_plus
tff(fact_6824_take__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : take(A,Nb,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),take(A,Nb,Xs)),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% take_append
tff(fact_6825_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Yb: 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),Xb),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Yb) = 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),Yb),Ys)) ).

% list_update_length
tff(fact_6826_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K2))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),K2))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K2),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_6827_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K2))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),K2))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K2)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_6828_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_6829_append__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Ys: list(A),Zs: list(A),F3: 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),map(B,A,F3),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),map(B,A,F3),Us2) )
          & ( Zs = aa(list(B),list(A),map(B,A,F3),Vs2) ) ) ) ).

% append_eq_map_conv
tff(fact_6830_map__eq__append__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),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),map(B,A,F3),Us2) )
          & ( Zs = aa(list(B),list(A),map(B,A,F3),Vs2) ) ) ) ).

% map_eq_append_conv
tff(fact_6831_replicate__app__Cons__same,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),Xs)) ).

% replicate_app_Cons_same
tff(fact_6832_split__list,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,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),Xb),Zs2)) ) ).

% split_list
tff(fact_6833_split__list__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,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),Xb),Zs2)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Zs2)) ) ) ).

% split_list_last
tff(fact_6834_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P2,X) )
     => ? [Ys3: list(A),X3: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
          & aa(A,$o,P2,X3) ) ) ).

% split_list_prop
tff(fact_6835_split__list__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,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),Xb),Zs2)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys3)) ) ) ).

% split_list_first
tff(fact_6836_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P2,X) )
     => ~ ! [Ys3: list(A),X3: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
           => ~ aa(A,$o,P2,X3) ) ) ).

% split_list_propE
tff(fact_6837_append__Cons__eq__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),Xs5: list(A),Ys6: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ member(A,Xb,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),Xb),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),Xb),Ys6)) )
        <=> ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_6838_in__set__conv__decomp,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,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),Xb),Zs3)) ) ).

% in_set_conv_decomp
tff(fact_6839_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P2,X) )
     => ? [Ys3: list(A),X3: A,Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2)) )
          & aa(A,$o,P2,X3)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
             => ~ aa(A,$o,P2,Xa) ) ) ) ).

% split_list_last_prop
tff(fact_6840_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P2,X) )
     => ? [Ys3: list(A),X3: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
          & aa(A,$o,P2,X3)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
             => ~ aa(A,$o,P2,Xa) ) ) ) ).

% split_list_first_prop
tff(fact_6841_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P2,X) )
     => ~ ! [Ys3: list(A),X3: A,Zs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2)) )
           => ( aa(A,$o,P2,X3)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
                   => ~ aa(A,$o,P2,Xa) ) ) ) ) ).

% split_list_last_propE
tff(fact_6842_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P2,X) )
     => ~ ! [Ys3: list(A),X3: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
           => ( aa(A,$o,P2,X3)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
                   => ~ aa(A,$o,P2,Xa) ) ) ) ) ).

% split_list_first_propE
tff(fact_6843_in__set__conv__decomp__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,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),Xb),Zs3)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Zs3)) ) ) ).

% in_set_conv_decomp_last
tff(fact_6844_in__set__conv__decomp__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,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),Xb),Zs3)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys4)) ) ) ).

% in_set_conv_decomp_first
tff(fact_6845_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P2,X4) )
    <=> ? [Ys4: list(A),X4: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs3)) )
          & aa(A,$o,P2,X4)
          & ! [Xa2: A] :
              ( member(A,Xa2,aa(list(A),set(A),set2(A),Zs3))
             => ~ aa(A,$o,P2,Xa2) ) ) ) ).

% split_list_last_prop_iff
tff(fact_6846_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P2,X4) )
    <=> ? [Ys4: list(A),X4: A] :
          ( ? [Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs3))
          & aa(A,$o,P2,X4)
          & ! [Xa2: A] :
              ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys4))
             => ~ aa(A,$o,P2,Xa2) ) ) ) ).

% split_list_first_prop_iff
tff(fact_6847_Cons__eq__appendI,axiom,
    ! [A: $tType,Xb: 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),Xb),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),Xb),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) ) ) ) ).

% Cons_eq_appendI
tff(fact_6848_append__Cons,axiom,
    ! [A: $tType,Xb: 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),Xb),Xs)),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% append_Cons
tff(fact_6849_concat_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: list(A),Xs: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xb),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),concat(A,Xs)) ).

% concat.simps(2)
tff(fact_6850_concat__eq__append__conv,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( concat(A,Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> $ite(
          Xss = nil(list(A)),
          ( ( Ys = nil(A) )
          & ( Zs = nil(A) ) ),
          ? [Xss1: list(list(A)),Xs3: list(A),Xs6: 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),Xs3),Xs6)),Xss22)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xss1)),Xs3) )
            & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs6),concat(A,Xss22)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_6851_concat__eq__appendD,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs: 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)) )
       => ? [Xss12: list(list(A)),Xs2: list(A),Xs4: 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),Xs2),Xs4)),Xss23)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xss12)),Xs2) )
            & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),concat(A,Xss23)) ) ) ) ) ).

% concat_eq_appendD
tff(fact_6852_rev__induct,axiom,
    ! [A: $tType,P2: fun(list(A),$o),Xs: list(A)] :
      ( aa(list(A),$o,P2,nil(A))
     => ( ! [X3: A,Xs2: list(A)] :
            ( aa(list(A),$o,P2,Xs2)
           => aa(list(A),$o,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))) )
       => aa(list(A),$o,P2,Xs) ) ) ).

% rev_induct
tff(fact_6853_rev__exhaust,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ~ ! [Ys3: list(A),Y: 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),Y),nil(A))) ) ).

% rev_exhaust
tff(fact_6854_Cons__eq__append__conv,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),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),Xb),Xs) = Zs ) )
        | ? [Ys7: list(A)] :
            ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),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_6855_append__eq__Cons__conv,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),Xb: 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),Xb),Xs) )
    <=> ( ( ( Ys = nil(A) )
          & ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) ) )
        | ? [Ys7: list(A)] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),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_6856_rev__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(list(A),$o)] :
      ( ( Xs != nil(A) )
     => ( ! [X3: A] : aa(list(A),$o,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ( ! [X3: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P2,Xs2)
               => aa(list(A),$o,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))) ) )
         => aa(list(A),$o,P2,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_6857_list__encode_Ocases,axiom,
    ! [Xb: list(nat)] :
      ( ( Xb != nil(nat) )
     => ~ ! [X3: nat,Xs2: list(nat)] : Xb != aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2) ) ).

% list_encode.cases
tff(fact_6858_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),R2: set(product_prod(A,A)),Xs: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R2))
     => 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_6859_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V: list(A),R2: set(product_prod(A,A)),Xb: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V),lexord(A,R2))
     => 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),Xb),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),V)),lexord(A,R2)) ) ).

% lexord_append_leftI
tff(fact_6860_append__listrel1I,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
      ( ( ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Vs),listrel1(A,R2)) ) )
     => 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_6861_append__replicate__commute,axiom,
    ! [A: $tType,Nb: nat,Xb: A,K2: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),replicate(A,K2,Xb)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,K2,Xb)),replicate(A,Nb,Xb)) ).

% append_replicate_commute
tff(fact_6862_replicate__add,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xb: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb),Xb) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),replicate(A,Mb,Xb)) ).

% replicate_add
tff(fact_6863_remove1__append,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] :
      remove1(A,Xb,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remove1(A,Xb,Xs)),Ys),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),remove1(A,Xb,Ys))) ).

% remove1_append
tff(fact_6864_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_6865_enumerate__append__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : enumerate(A,Nb,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,Nb,Xs)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_6866_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_6867_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_6868_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_6869_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_6870_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_6871_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) )
     => ? [M: nat,N2: nat,Zs2: list(A)] :
          ( ( concat(A,replicate(list(A),M,Zs2)) = Xs )
          & ( concat(A,replicate(list(A),N2,Zs2)) = Ys ) ) ) ).

% comm_append_are_replicate
tff(fact_6872_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),X3: A,Xs4: list(A),Y: A,Ys5: list(A)] :
            ( ( X3 != Y )
            & ( 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),X3),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),Y),nil(A))),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_6873_not__distinct__decomp,axiom,
    ! [A: $tType,Ws2: list(A)] :
      ( ~ distinct(A,Ws2)
     => ? [Xs2: list(A),Ys3: list(A),Zs2: list(A),Y: 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),Y),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),Y),nil(A))),Zs2)))) ) ).

% not_distinct_decomp
tff(fact_6874_not__distinct__conv__prefix,axiom,
    ! [A: $tType,As: list(A)] :
      ( ~ distinct(A,As)
    <=> ? [Xs3: list(A),Y4: A,Ys4: list(A)] :
          ( 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_6875_replicate__append__same,axiom,
    ! [A: $tType,I2: nat,Xb: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,I2,Xb)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),replicate(A,I2,Xb)) ).

% replicate_append_same
tff(fact_6876_list__update__append1,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),I2,Xb) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,I2,Xb)),Ys) ) ) ).

% list_update_append1
tff(fact_6877_remdups__adj__append__two,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Yb: 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),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),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),Xb),nil(A))))),
        $ite(Xb = Yb,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),nil(A)))) ).

% remdups_adj_append_two
tff(fact_6878_remove1__split,axiom,
    ! [A: $tType,A2: A,Xs: list(A),Ys: list(A)] :
      ( 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)) )
            & ~ 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_6879_lexord__append__leftD,axiom,
    ! [A: $tType,Xb: list(A),U: list(A),V: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),V)),lexord(A,R2))
     => ( ! [A4: A] : ~ 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)
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V),lexord(A,R2)) ) ) ).

% lexord_append_leftD
tff(fact_6880_lexord__append__rightI,axiom,
    ! [A: $tType,Yb: list(A),Xb: list(A),R2: set(product_prod(A,A))] :
      ( ? [B9: A,Z3: list(A)] : Yb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B9),Z3)
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),Yb)),lexord(A,R2)) ) ).

% lexord_append_rightI
tff(fact_6881_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(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) )
           => 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_6882_lex__append__left__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ 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)
     => ( 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))
      <=> 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_6883_lex__append__leftD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ 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)
     => ( 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))
       => 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_6884_rotate1_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: 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),Xb),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),Xb),nil(A))) ).

% rotate1.simps(2)
tff(fact_6885_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),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_6886_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs),lenlex(A,R))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),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_6887_nths__append,axiom,
    ! [A: $tType,L: list(A),L2: list(A),A5: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L2),A5) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A5)),nths(A,L2,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_aem(list(A),fun(set(nat),fun(nat,$o)),L),A5)))) ).

% nths_append
tff(fact_6888_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [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) = Nb ) ) ) ).

% length_Suc_conv_rev
tff(fact_6889_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      subseqs(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $let(
        xss: list(list(A)),
        xss:= subseqs(A,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)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb)),xss)),xss) ) ).

% subseqs.simps(2)
tff(fact_6890_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,A,nth(A,Xs),Nb),aa(nat,A,nth(A,Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)))) ).

% nth_append
tff(fact_6891_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat,Xb: A] :
      list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),Nb,Xb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,Nb,Xb)),Ys),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),list_update(A,Ys,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Xb))) ).

% list_update_append
tff(fact_6892_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
     => ~ ! [X3: A,Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),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),X3),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),Y),Vs3)) ) ) ) ) ).

% listrel1E
tff(fact_6893_listrel1I,axiom,
    ! [A: $tType,Xb: A,Yb: A,R2: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),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),Xb),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),Yb),Vs)) )
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2)) ) ) ) ).

% listrel1I
tff(fact_6894_lexord__append__left__rightI,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),U: list(A),Xb: list(A),Yb: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
     => 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),Xb))),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),Yb))),lexord(A,R2)) ) ).

% lexord_append_left_rightI
tff(fact_6895_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(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))
    <=> ( ? [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2) )
        | 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_6896_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R2: set(product_prod(A,A)),V: list(A),Z: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W2),lexord(A,R2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),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),V)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),Z)),lexord(A,R2)) ) ) ).

% lexord_sufI
tff(fact_6897_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = 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)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb)),Ys)),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_6898_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Yb: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),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),Yb),nil(A)))),listrel1(A,R2))
    <=> ( ( 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))
          & ( Xb = Yb ) )
        | ( ( Xs = Ys )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_6899_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : groups4207007520872428315er_sum(B,A,F3,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),groups4207007520872428315er_sum(B,A,F3,A2,Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(list(B),nat,size_size(list(B)),Xs))),groups4207007520872428315er_sum(B,A,F3,A2,Ys))) ) ).

% horner_sum_append
tff(fact_6900_nths__Cons,axiom,
    ! [A: $tType,Xb: A,L: list(A),A5: set(nat)] :
      nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),L),A5) = aa(list(A),list(A),
        aa(list(A),fun(list(A),list(A)),append(A),
          $ite(member(nat,zero_zero(nat),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)),nil(A))),
        nths(A,L,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_aen(set(nat),fun(nat,$o),A5)))) ).

% nths_Cons
tff(fact_6901_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N2: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N2)
              & ( concat(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_6902_listrel1__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_aeo(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% listrel1_def
tff(fact_6903_lexord__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_aep(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lexord_def
tff(fact_6904_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,aa(nat,nat,suc,I2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_6905_lex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_aeq(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lex_conv
tff(fact_6906_nth__repl,axiom,
    ! [A: $tType,Mb: nat,Xs: list(A),Nb: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( Mb != Nb )
         => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,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),Xb),nil(A))),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),Xs)))),Mb) = aa(nat,A,nth(A,Xs),Mb) ) ) ) ) ).

% nth_repl
tff(fact_6907_pos__n__replace,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Yb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,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),Yb),nil(A))),drop(A,aa(nat,nat,suc,Nb),Xs)))) ) ) ).

% pos_n_replace
tff(fact_6908_drop0,axiom,
    ! [A: $tType,X: list(A)] : drop(A,zero_zero(nat),X) = X ).

% drop0
tff(fact_6909_drop__drop,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] : drop(A,Nb,drop(A,Mb,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb),Xs) ).

% drop_drop
tff(fact_6910_drop__Suc__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,Nb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = drop(A,Nb,Xs) ).

% drop_Suc_Cons
tff(fact_6911_length__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),drop(A,Nb,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_drop
tff(fact_6912_drop__update__cancel,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( drop(A,Mb,list_update(A,Xs,Nb,Xb)) = drop(A,Mb,Xs) ) ) ).

% drop_update_cancel
tff(fact_6913_append__take__drop__id,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,Xs)),drop(A,Nb,Xs)) = Xs ).

% append_take_drop_id
tff(fact_6914_drop__replicate,axiom,
    ! [A: $tType,I2: nat,K2: nat,Xb: A] : drop(A,I2,replicate(A,K2,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),I2),Xb) ).

% drop_replicate
tff(fact_6915_drop__eq__Nil2,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,Nb,Xs) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% drop_eq_Nil2
tff(fact_6916_drop__eq__Nil,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( drop(A,Nb,Xs) = nil(A) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% drop_eq_Nil
tff(fact_6917_drop__all,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)
     => ( drop(A,Nb,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_6918_drop__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : drop(A,Nb,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),drop(A,Nb,Xs)),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% drop_append
tff(fact_6919_drop__Cons__numeral,axiom,
    ! [A: $tType,V: num,Xb: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_6920_nth__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,drop(A,Nb,Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),I2)) ) ) ).

% nth_drop
tff(fact_6921_nth__via__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Yb: A,Ys: list(A)] :
      ( ( drop(A,Nb,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys) )
     => ( aa(nat,A,nth(A,Xs),Nb) = Yb ) ) ).

% nth_via_drop
tff(fact_6922_drop__zip,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] : drop(product_prod(A,B),Nb,zip(A,B,Xs,Ys)) = zip(A,B,drop(A,Nb,Xs),drop(B,Nb,Ys)) ).

% drop_zip
tff(fact_6923_drop__Nil,axiom,
    ! [A: $tType,Nb: nat] : drop(A,Nb,nil(A)) = nil(A) ).

% drop_Nil
tff(fact_6924_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_6925_take__drop,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] : take(A,Nb,drop(A,Mb,Xs)) = drop(A,Mb,take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb),Xs)) ).

% take_drop
tff(fact_6926_in__set__dropD,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),drop(A,Nb,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_dropD
tff(fact_6927_drop__take,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] : drop(A,Nb,take(A,Mb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),drop(A,Nb,Xs)) ).

% drop_take
tff(fact_6928_set__drop__subset,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Nb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_drop_subset
tff(fact_6929_distinct__drop,axiom,
    ! [A: $tType,Xs: list(A),I2: nat] :
      ( distinct(A,Xs)
     => distinct(A,drop(A,I2,Xs)) ) ).

% distinct_drop
tff(fact_6930_drop__eq__nths,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,Nb,Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),Nb))) ).

% drop_eq_nths
tff(fact_6931_drop__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F3: fun(B,A),Xs: list(B)] : drop(A,Nb,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),drop(B,Nb,Xs)) ).

% drop_map
tff(fact_6932_set__drop__subset__set__drop,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Mb,Xs))),aa(list(A),set(A),set2(A),drop(A,Nb,Xs))) ) ).

% set_drop_subset_set_drop
tff(fact_6933_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 = take(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) )
        & ( Ys = drop(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_6934_take__add,axiom,
    ! [A: $tType,I2: nat,J: nat,Xs: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),take(A,J,drop(A,I2,Xs))) ).

% take_add
tff(fact_6935_drop__update__swap,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( drop(A,Mb,list_update(A,Xs,Nb,Xb)) = list_update(A,drop(A,Mb,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb),Xb) ) ) ).

% drop_update_swap
tff(fact_6936_drop__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : drop(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aTP_Lamp_aer(list(A),fun(nat,list(A)),Xs),Nb) ).

% drop_Cons
tff(fact_6937_nths__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I6: set(nat)] : nths(A,drop(A,Nb,Xs),I6) = nths(A,Xs,aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb)),I6)) ).

% nths_drop
tff(fact_6938_drop__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      drop(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(Nb = zero_zero(nat),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs)) ).

% drop_Cons'
tff(fact_6939_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) )
    <=> $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)),
          ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
          & ( Xs_2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1)),Ys_2) ) ),
          ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
          & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1)),Xs_2) = Ys_2 ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_6940_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,take(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))),zip(A,B,Ys,drop(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))) ).

% zip_append1
tff(fact_6941_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,take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Ys)),zip(A,B,drop(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Zs)) ).

% zip_append2
tff(fact_6942_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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),I2)),drop(A,aa(nat,nat,suc,I2),Xs)) = drop(A,I2,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_6943_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I2: nat,J: nat] :
      ( distinct(A,Vs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I2,Vs))),aa(list(A),set(A),set2(A),drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_6944_id__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_6945_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),A2: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,Xs,I2,A2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_6946_lexn__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Nb: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),Nb) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_aes(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),Nb))) ).

% lexn_conv
tff(fact_6947_list__encode_Oelims,axiom,
    ! [Xb: list(nat),Yb: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = Yb )
     => ( ( ( Xb = nil(nat) )
         => ( Yb != zero_zero(nat) ) )
       => ~ ! [X3: nat,Xs2: list(nat)] :
              ( ( Xb = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2) )
             => ( Yb != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),aa(list(nat),nat,nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_6948_inj__list__encode,axiom,
    ! [A5: set(list(nat))] : inj_on(list(nat),nat,nat_list_encode,A5) ).

% inj_list_encode
tff(fact_6949_surj__list__encode,axiom,
    aa(set(list(nat)),set(nat),image2(list(nat),nat,nat_list_encode),top_top(set(list(nat)))) = top_top(set(nat)) ).

% surj_list_encode
tff(fact_6950_list__encode__eq,axiom,
    ! [Xb: list(nat),Yb: list(nat)] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = aa(list(nat),nat,nat_list_encode,Yb) )
    <=> ( Xb = Yb ) ) ).

% list_encode_eq
tff(fact_6951_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_6952_list__encode_Osimps_I1_J,axiom,
    aa(list(nat),nat,nat_list_encode,nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_6953_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Nb: nat] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),Nb))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = Nb ) ) ) ).

% lexn_length
tff(fact_6954_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_6955_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list(nat)] : aa(list(nat),nat,nat_list_encode,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Xb),Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),aa(list(nat),nat,nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_6956_upto__aux__rec,axiom,
    ! [I2: int,J: int,Js: list(int)] :
      upto_aux(I2,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I2),Js,upto_aux(I2,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_6957_list_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,fun(list(B),A)),X21: B,X222: list(B)] : aa(list(B),A,case_list(A,B,F1,F22),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X21),X222)) = aa(list(B),A,aa(B,fun(list(B),A),F22,X21),X222) ).

% list.simps(5)
tff(fact_6958_list_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,fun(list(C),B)),List: list(C)] : aa(B,A,H,aa(list(C),B,case_list(B,C,F1,F22),List)) = aa(list(C),A,case_list(A,C,aa(B,A,H,F1),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_aet(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),H),F22)),List) ).

% list.case_distrib
tff(fact_6959_list_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,fun(list(B),A))] : aa(list(B),A,case_list(A,B,F1,F22),nil(B)) = F1 ).

% list.simps(4)
tff(fact_6960_remdups__adj__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)),aTP_Lamp_aeu(A,fun(A,fun(list(A),list(A))),Xb)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_6961_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),A,case_list(A,A,Xb,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_aev(A,fun(list(A),fun(A,fun(list(A),A))),Xb),Xs)),Xs) ) ).

% min_list.simps
tff(fact_6962_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: 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),Xb),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),Xb),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aew(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),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aex(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_6963_transpose_Oelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Yb: list(list(A))] :
      ( ( transpose(A,Xb) = Yb )
     => ( ( ( Xb = nil(list(A)) )
         => ( Yb != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( Xb = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
             => ( Yb != transpose(A,Xss2) ) )
         => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( Xb = 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),X3),Xs2)),Xss2) )
               => ( Yb != 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),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aew(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),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aex(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_6964_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_6965_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> aa(list(A),$o,case_list($o,A,$false,aTP_Lamp_aey(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_6966_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> aa(list(A),$o,case_list($o,A,$true,aTP_Lamp_aez(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_6967_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).

% transpose.simps(1)
tff(fact_6968_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),Xs)) = aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),transpose(B,Xs)) ).

% transpose_map_map
tff(fact_6969_transpose__empty,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( transpose(A,Xs) = nil(list(A)) )
    <=> ! [X4: list(A)] :
          ( member(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( X4 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_6970_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = aa(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_afa(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Xb),Xs)),Ys) ).

% zip_Cons1
tff(fact_6971_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Yb: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Yb),Ys)) = aa(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_afb(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Yb),Ys)),Xs) ).

% zip_Cons
tff(fact_6972_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Yb: list(int)] :
      ( ( upto(Xb,Xaa) = Yb )
     => ( 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),Xb),Xaa))
       => ~ ( ( Yb = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Xb),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),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),Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_6973_upto_Opsimps,axiom,
    ! [I2: 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),I2),J))
     => ( upto(I2,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_6974_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( upto(I2,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I2) ) ).

% upto_Nil
tff(fact_6975_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil(int) = upto(I2,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I2) ) ).

% upto_Nil2
tff(fact_6976_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I2)
     => ( upto(I2,J) = nil(int) ) ) ).

% upto_empty
tff(fact_6977_upto__single,axiom,
    ! [I2: int] : upto(I2,I2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),nil(int)) ).

% upto_single
tff(fact_6978_nth__upto,axiom,
    ! [I2: int,K2: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K2))),J)
     => ( aa(nat,int,nth(int,upto(I2,J)),K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K2)) ) ) ).

% nth_upto
tff(fact_6979_length__upto,axiom,
    ! [I2: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I2,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),I2)),one_one(int))) ).

% length_upto
tff(fact_6980_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa(num,int,numeral_numeral(int),Mb),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Mb)),aa(num,int,numeral_numeral(int),Nb)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),Mb)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Mb)),one_one(int)),aa(num,int,numeral_numeral(int),Nb))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_6981_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa(num,int,numeral_numeral(int),Mb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),Mb)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Mb)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_6982_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb)),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),aa(num,int,numeral_numeral(int),Nb)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),one_one(int)),aa(num,int,numeral_numeral(int),Nb))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_6983_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_6984_atLeastAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or1337092689740270186AtMost(int,I2,J) = aa(list(int),set(int),set2(int),upto(I2,J)) ).

% atLeastAtMost_upto
tff(fact_6985_distinct__upto,axiom,
    ! [I2: int,J: int] : distinct(int,upto(I2,J)) ).

% distinct_upto
tff(fact_6986_upto__aux__def,axiom,
    ! [I2: int,J: int,Js: list(int)] : upto_aux(I2,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,J)),Js) ).

% upto_aux_def
tff(fact_6987_upto__code,axiom,
    ! [I2: int,J: int] : upto(I2,J) = upto_aux(I2,J,nil(int)) ).

% upto_code
tff(fact_6988_upto__split2,axiom,
    ! [I2: int,J: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K2)
       => ( upto(I2,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K2)) ) ) ) ).

% upto_split2
tff(fact_6989_upto__split1,axiom,
    ! [I2: int,J: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K2)
       => ( upto(I2,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K2)) ) ) ) ).

% upto_split1
tff(fact_6990_atLeastLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or7035219750837199246ssThan(int,I2,J) = aa(list(int),set(int),set2(int),upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6991_greaterThanAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or3652927894154168847AtMost(int,I2,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_6992_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6993_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Yb: list(int)] :
      ( ( upto(Xb,Xaa) = Yb )
     => ( Yb = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Xb),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_6994_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      upto(I2,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_6995_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( upto(I2,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,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_6996_greaterThanLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or5935395276787703475ssThan(int,I2,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_6997_upto__split3,axiom,
    ! [I2: int,J: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K2)
       => ( upto(I2,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,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)),K2))) ) ) ) ).

% upto_split3
tff(fact_6998_transpose_Opelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Yb: list(list(A))] :
      ( ( transpose(A,Xb) = Yb )
     => ( accp(list(list(A)),transpose_rel(A),Xb)
       => ( ( ( Xb = nil(list(A)) )
           => ( ( Yb = nil(list(A)) )
             => ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( Xb = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
               => ( ( Yb = 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)) ) )
           => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( Xb = 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),X3),Xs2)),Xss2) )
                 => ( ( Yb = 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),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aew(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),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aex(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),X3),Xs2)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_6999_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,Xb: 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),Xb),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),Xb),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),Xb),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aew(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),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aex(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_7000_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_7001_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_7002_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P2: fun(list(list(A)),$o)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => aa(list(list(A)),$o,P2,nil(list(A))) )
       => ( ! [Xss2: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))
             => ( aa(list(list(A)),$o,P2,Xss2)
               => aa(list(list(A)),$o,P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) ) )
         => ( ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))
               => ( aa(list(list(A)),$o,P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aex(A,fun(list(A),list(list(A)))))),Xss2))))
                 => aa(list(list(A)),$o,P2,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2)) ) )
           => aa(list(list(A)),$o,P2,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_7003_remdups__adj_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Yb: list(A)] :
      ( ( remdups_adj(A,Xb) = Yb )
     => ( accp(list(A),remdups_adj_rel(A),Xb)
       => ( ( ( Xb = nil(A) )
           => ( ( Yb = nil(A) )
             => ~ accp(list(A),remdups_adj_rel(A),nil(A)) ) )
         => ( ! [X3: A] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
               => ( ( Yb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
                 => ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))) ) )
           => ~ ! [X3: A,Y: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2)) )
                 => ( ( Yb = $ite(X3 = Y,remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2)))) )
                   => ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2))) ) ) ) ) ) ) ).

% remdups_adj.pelims
tff(fact_7004_take__hd__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),drop(A,Nb,Xs))),nil(A))) = take(A,aa(nat,nat,suc,Nb),Xs) ) ) ).

% take_hd_drop
tff(fact_7005_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_7006_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_7007_hd__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),replicate(A,Nb,Xb)) = Xb ) ) ).

% hd_replicate
tff(fact_7008_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),J)
     => ( aa(list(A),A,hd(A),take(A,J,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_take
tff(fact_7009_list_Oset__sel_I1_J,axiom,
    ! [A: $tType,A2: list(A)] :
      ( ( A2 != nil(A) )
     => member(A,aa(list(A),A,hd(A),A2),aa(list(A),set(A),set2(A),A2)) ) ).

% list.set_sel(1)
tff(fact_7010_hd__in__set,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => member(A,aa(list(A),A,hd(A),Xs),aa(list(A),set(A),set2(A),Xs)) ) ).

% hd_in_set
tff(fact_7011_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_7012_hd__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(Xs = nil(A),aa(list(A),A,hd(A),Ys),aa(list(A),A,hd(A),Xs)) ).

% hd_append
tff(fact_7013_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_7014_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),concat(A,Xs)) = aa(list(A),A,hd(A),aa(list(list(A)),list(A),hd(list(A)),Xs)) ) ) ) ).

% hd_concat
tff(fact_7015_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_7016_list_Omap__sel_I1_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F3: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F3),A2)) = aa(A,B,F3,aa(list(A),A,hd(A),A2)) ) ) ).

% list.map_sel(1)
tff(fact_7017_hd__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,B)] :
      ( ( Xs != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F3),Xs)) = aa(A,B,F3,aa(list(A),A,hd(A),Xs)) ) ) ).

% hd_map
tff(fact_7018_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_7019_hd__drop__conv__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),A,hd(A),drop(A,Nb,Xs)) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% hd_drop_conv_nth
tff(fact_7020_remdups__adj__singleton__iff,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Xs != nil(A) )
        & ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),aa(list(A),A,hd(A),Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_7021_list__encode_Opelims,axiom,
    ! [Xb: list(nat),Yb: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = Yb )
     => ( accp(list(nat),nat_list_encode_rel,Xb)
       => ( ( ( Xb = nil(nat) )
           => ( ( Yb = zero_zero(nat) )
             => ~ accp(list(nat),nat_list_encode_rel,nil(nat)) ) )
         => ~ ! [X3: nat,Xs2: list(nat)] :
                ( ( Xb = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2) )
               => ( ( Yb = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),aa(list(nat),nat,nat_list_encode,Xs2)))) )
                 => ~ accp(list(nat),nat_list_encode_rel,aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_7022_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list(nat),list($o),map(nat,$o,bit_se5641148757651400278ts_bit(A,A2)),upt(zero_zero(nat),Nb))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_7023_remdups__upt,axiom,
    ! [Mb: nat,Nb: nat] : remdups(nat,upt(Mb,Nb)) = upt(Mb,Nb) ).

% remdups_upt
tff(fact_7024_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(list(nat),nat,hd(nat),upt(I2,J)) = I2 ) ) ).

% hd_upt
tff(fact_7025_drop__upt,axiom,
    ! [Mb: nat,I2: nat,J: nat] : drop(nat,Mb,upt(I2,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Mb),J) ).

% drop_upt
tff(fact_7026_length__upt,axiom,
    ! [I2: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I2,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2) ).

% length_upt
tff(fact_7027_take__upt,axiom,
    ! [I2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Mb)),Nb)
     => ( take(nat,Mb,upt(I2,Nb)) = upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Mb)) ) ) ).

% take_upt
tff(fact_7028_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( upt(I2,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_7029_sorted__list__of__set__range,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or7035219750837199246ssThan(nat,Mb,Nb)) = upt(Mb,Nb) ).

% sorted_list_of_set_range
tff(fact_7030_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( upt(I2,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2) ) ) ).

% upt_eq_Nil_conv
tff(fact_7031_nth__upt,axiom,
    ! [I2: nat,K2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2)),J)
     => ( aa(nat,nat,nth(nat,upt(I2,J)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2) ) ) ).

% nth_upt
tff(fact_7032_map__fst__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Nb,Xs)) = upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_7033_upt__rec__numeral,axiom,
    ! [Mb: num,Nb: num] :
      upt(aa(num,nat,numeral_numeral(nat),Mb),aa(num,nat,numeral_numeral(nat),Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),Mb)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb))),nil(nat)) ).

% upt_rec_numeral
tff(fact_7034_upt__0,axiom,
    ! [I2: nat] : upt(I2,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_7035_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Nb: nat,Ns: list(nat),Q2: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Mb),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Nb),Ns)) = upt(Mb,Q2) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Nb),Ns) = upt(aa(nat,nat,suc,Mb),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_7036_enumerate__map__upt,axiom,
    ! [A: $tType,Nb: nat,F3: fun(nat,A),Mb: nat] : enumerate(A,Nb,aa(list(nat),list(A),map(nat,A,F3),upt(Nb,Mb))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_afc(fun(nat,A),fun(nat,product_prod(nat,A)),F3)),upt(Nb,Mb)) ).

% enumerate_map_upt
tff(fact_7037_map__Suc__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa(list(nat),list(nat),map(nat,nat,suc),upt(Mb,Nb)) = upt(aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb)) ).

% map_Suc_upt
tff(fact_7038_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_afd(nat,fun(nat,nat),Nb)),upt(zero_zero(nat),Mb)) = upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% map_add_upt
tff(fact_7039_map__replicate__trivial,axiom,
    ! [A: $tType,Xb: A,I2: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_afe(A,fun(nat,A),Xb)),upt(zero_zero(nat),I2)) = replicate(A,I2,Xb) ).

% map_replicate_trivial
tff(fact_7040_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or5935395276787703475ssThan(nat,Nb,Mb) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Nb),Mb)) ).

% greaterThanLessThan_upt
tff(fact_7041_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or3652927894154168847AtMost(nat,Nb,Mb) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Nb),aa(nat,nat,suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_7042_atLeast__upt,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_lessThan(nat),Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Nb)) ).

% atLeast_upt
tff(fact_7043_atLeastLessThan__upt,axiom,
    ! [I2: nat,J: nat] : set_or7035219750837199246ssThan(nat,I2,J) = aa(list(nat),set(nat),set2(nat),upt(I2,J)) ).

% atLeastLessThan_upt
tff(fact_7044_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or1337092689740270186AtMost(nat,Nb,Mb) = aa(list(nat),set(nat),set2(nat),upt(Nb,aa(nat,nat,suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_7045_distinct__upt,axiom,
    ! [I2: nat,J: nat] : distinct(nat,upt(I2,J)) ).

% distinct_upt
tff(fact_7046_atMost__upto,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_atMost(nat),Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) ).

% atMost_upto
tff(fact_7047_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J)) ) ) ).

% upt_conv_Cons
tff(fact_7048_enumerate__eq__zip,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : enumerate(A,Nb,Xs) = zip(nat,A,upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_7049_map__upt__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),Nb: nat] : aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F3,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_aff(fun(nat,A),fun(nat,A),F3)),upt(zero_zero(nat),Nb))) ).

% map_upt_Suc
tff(fact_7050_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_yf(nat,nat)),upt(aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_7051_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(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_7052_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2))) ) ) ).

% upt_add_eq_append
tff(fact_7053_nth__map__upt,axiom,
    ! [A: $tType,I2: nat,Nb: nat,Mb: nat,F3: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F3),upt(Mb,Nb))),I2) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I2)) ) ) ).

% nth_map_upt
tff(fact_7054_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,Xb: nat,Xs: list(nat)] :
      ( ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Xb),Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
        & ( I2 = Xb )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_7055_upt__rec,axiom,
    ! [I2: nat,J: nat] :
      upt(I2,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J)),nil(nat)) ).

% upt_rec
tff(fact_7056_enumerate__replicate__eq,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,A2: A] : enumerate(A,Nb,replicate(A,Mb,A2)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_afg(A,fun(nat,product_prod(nat,A)),A2)),upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb))) ).

% enumerate_replicate_eq
tff(fact_7057_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Mb: nat,F3: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I3)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F3),upt(Mb,Nb)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_7058_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      upt(I2,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J),aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_7059_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( upt(I2,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_7060_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( Nb = zero_zero(nat) ) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I3)) = Nb ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_afi(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),Nb)) ) ) ) ).

% transpose_rectangle
tff(fact_7061_extract__Some__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Ys: list(A),Yb: A,Zs: list(A)] :
      ( ( extract(A,P2,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)),Yb),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),Yb),Zs)) )
        & aa(A,$o,P2,Yb)
        & ~ ? [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P2,X4) ) ) ) ).

% extract_Some_iff
tff(fact_7062_extract__SomeE,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Ys: list(A),Yb: A,Zs: list(A)] :
      ( ( extract(A,P2,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)),Yb),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),Yb),Zs)) )
        & aa(A,$o,P2,Yb)
        & ~ ? [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P2,X) ) ) ) ).

% extract_SomeE
tff(fact_7063_extract__None__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( extract(A,P2,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P2,X4) ) ) ).

% extract_None_iff
tff(fact_7064_extract__Nil__code,axiom,
    ! [A: $tType,P2: fun(A,$o)] : extract(A,P2,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ).

% extract_Nil_code
tff(fact_7065_extract__Cons__code,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A)] :
      extract(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P2,Xb),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)),Xb),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_afk(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Xb)),extract(A,P2,Xs))) ).

% extract_Cons_code
tff(fact_7066_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Nb: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),aa(nat,nat,suc,Nb)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),aa(set(product_prod(product_prod(A,list(A)),product_prod(A,list(A)))),set(product_prod(list(A),list(A))),image2(product_prod(product_prod(A,list(A)),product_prod(A,list(A))),product_prod(list(A),list(A)),product_map_prod(product_prod(A,list(A)),list(A),product_prod(A,list(A)),list(A),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)))),lex_prod(A,list(A),R2,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),Nb)))),aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_afl(nat,fun(list(A),fun(list(A),$o)),Nb)))) ).

% lexn.simps(2)
tff(fact_7067_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),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_afm(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F3)),X5) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_7068_map__prod__ident,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_adm(A,A),aTP_Lamp_afn(B,B)),X) = X ).

% map_prod_ident
tff(fact_7069_map__prod__simp,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A2: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),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,F3,A2)),aa(D,B,G3,B2)) ).

% map_prod_simp
tff(fact_7070_fst__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),Xb: product_prod(C,D)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),Xb)) = aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),Xb)) ).

% fst_map_prod
tff(fact_7071_snd__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(C,B),G3: fun(D,A),Xb: product_prod(C,D)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,D),product_prod(B,A),product_map_prod(C,B,D,A,F3,G3),Xb)) = aa(D,A,G3,aa(product_prod(C,D),D,product_snd(C,D),Xb)) ).

% snd_map_prod
tff(fact_7072_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A,nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_7073_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( groups8242544230860333062m_list(A,Ns) = zero_zero(A) )
        <=> ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Ns))
             => ( X4 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_7074_sum__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xb: A,Xs: list(A)] : groups8242544230860333062m_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),groups8242544230860333062m_list(A,Xs)) ) ).

% sum_list.Cons
tff(fact_7075_sum__list__append,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A),Ys: list(A)] : groups8242544230860333062m_list(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),groups8242544230860333062m_list(A,Ys)) ) ).

% sum_list_append
tff(fact_7076_map__prod__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R: set(product_prod(A,B)),F3: fun(A,C),G3: fun(B,D)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),R)
     => 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,F3,A2)),aa(B,D,G3,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,F3,G3)),R)) ) ).

% map_prod_imageI
tff(fact_7077_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_afo(B,A)),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_7078_fst__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F3: fun(A,C),G3: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C),comp(product_prod(C,D),C,product_prod(A,B),product_fst(C,D)),product_map_prod(A,C,B,D,F3,G3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_map_prod
tff(fact_7079_snd__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F3: fun(A,D),G3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C),comp(product_prod(D,C),C,product_prod(A,B),product_snd(D,C)),product_map_prod(A,D,B,C,F3,G3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),G3),product_snd(A,B)) ).

% snd_comp_map_prod
tff(fact_7080_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( groups8242544230860333062m_list(nat,upt(Mb,Nb)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ca(nat,nat)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ) ).

% sum_list_upt
tff(fact_7081_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bs(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G3),Xs))) ) ).

% sum_list_addf
tff(fact_7082_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Xs: list(A)] :
          ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),groups8242544230860333062m_list(A,Xs)) ) ) ).

% member_le_sum_list
tff(fact_7083_map__prod__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: fun(A,C),G3: fun(B,D)] : product_map_prod(A,C,B,D,F3,G3) = 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_afp(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F3),G3)) ).

% map_prod_def
tff(fact_7084_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod(A,B),F3: fun(C,A),G3: fun(D,B),R: set(product_prod(C,D))] :
      ( member(product_prod(A,B),C3,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,F3,G3)),R))
     => ~ ! [X3: C,Y: D] :
            ( ( C3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X3)),aa(D,B,G3,Y)) )
           => ~ member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X3),Y),R) ) ) ).

% prod_fun_imageE
tff(fact_7085_map__prod_Ocomp,axiom,
    ! [A: $tType,E: $tType,C: $tType,D: $tType,F: $tType,B: $tType,F3: fun(E,C),G3: fun(F,D),H: fun(A,E),I2: fun(B,F)] : aa(fun(product_prod(A,B),product_prod(E,F)),fun(product_prod(A,B),product_prod(C,D)),comp(product_prod(E,F),product_prod(C,D),product_prod(A,B),product_map_prod(E,C,F,D,F3,G3)),product_map_prod(A,E,B,F,H,I2)) = product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),comp(E,C,A,F3),H),aa(fun(B,F),fun(B,D),comp(F,D,B,G3),I2)) ).

% map_prod.comp
tff(fact_7086_map__prod_Ocompositionality,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F: $tType,E: $tType,F3: fun(C,A),G3: fun(D,B),H: fun(E,C),I2: fun(F,D),Prod: product_prod(E,F)] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F3,G3),aa(product_prod(E,F),product_prod(C,D),product_map_prod(E,C,F,D,H,I2),Prod)) = aa(product_prod(E,F),product_prod(A,B),product_map_prod(E,A,F,B,aa(fun(E,C),fun(E,A),comp(C,A,E,F3),H),aa(fun(F,D),fun(F,B),comp(D,B,F,G3),I2)),Prod) ).

% map_prod.compositionality
tff(fact_7087_map__prod__compose,axiom,
    ! [D: $tType,C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,F1: fun(E,C),F22: fun(A,E),G1: fun(F,D),G22: fun(B,F)] : product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),comp(E,C,A,F1),F22),aa(fun(B,F),fun(B,D),comp(F,D,B,G1),G22)) = aa(fun(product_prod(A,B),product_prod(E,F)),fun(product_prod(A,B),product_prod(C,D)),comp(product_prod(E,F),product_prod(C,D),product_prod(A,B),product_map_prod(E,C,F,D,F1,G1)),product_map_prod(A,E,B,F,F22,G22)) ).

% map_prod_compose
tff(fact_7088_map__prod_Oidentity,axiom,
    ! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_adm(A,A),aTP_Lamp_afn(B,B)) = id(product_prod(A,B)) ).

% map_prod.identity
tff(fact_7089_apfst__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,C)] : product_apfst(A,C,B,F3) = product_map_prod(A,C,B,B,F3,id(B)) ).

% apfst_def
tff(fact_7090_apsnd__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] : product_apsnd(B,C,A,F3) = product_map_prod(A,A,B,C,id(A),F3) ).

% apsnd_def
tff(fact_7091_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),zero_zero(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),groups8242544230860333062m_list(A,Xs)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_7092_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X3) )
         => ( ( groups8242544230860333062m_list(A,Xs) = zero_zero(A) )
          <=> ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_7093_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),groups8242544230860333062m_list(A,Xs)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_7094_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xs))),groups8242544230860333062m_list(A,aa(list(A),list(A),map(A,A,abs_abs(A)),Xs))) ) ).

% sum_list_abs
tff(fact_7095_map__prod__surj,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,F3: fun(B,A),G3: fun(D,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => ( ( aa(set(D),set(C),image2(D,C,G3),top_top(set(D))) = top_top(set(C)) )
       => ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F3,G3)),top_top(set(product_prod(B,D)))) = top_top(set(product_prod(A,C))) ) ) ) ).

% map_prod_surj
tff(fact_7096_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs))) ) ) ).

% sum_list_mono
tff(fact_7097_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(A,Xs) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_afq(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_7098_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs))) ) ) ) ).

% sum_list_strict_mono
tff(fact_7099_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K2: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K2)),groups8242544230860333062m_list(A,Ns)) ) ) ).

% elem_le_sum_list
tff(fact_7100_sum_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(A),G3: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(list(A),set(A),set2(A),Xs)) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs)) ) ) ) ).

% sum.distinct_set_conv_list
tff(fact_7101_sum__list__distinct__conv__sum__set,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(A),F3: fun(A,B)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% sum_list_distinct_conv_sum_set
tff(fact_7102_sum__list__map__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xb: A,Xs: list(A),F3: fun(A,B)] :
          ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,Xb)),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),remove1(A,Xb,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_7103_sum__code,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),aa(list(B),set(B),set2(B),Xs)) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G3),remdups(B,Xs))) ) ).

% sum_code
tff(fact_7104_sum__set__upto__conv__sum__list__int,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(int,A),I2: int,J: int] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),F3),aa(list(int),set(int),set2(int),upto(I2,J))) = groups8242544230860333062m_list(A,aa(list(int),list(A),map(int,A,F3),upto(I2,J))) ) ).

% sum_set_upto_conv_sum_list_int
tff(fact_7105_interv__sum__list__conv__sum__set__int,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(int,A),K2: int,L: int] : groups8242544230860333062m_list(A,aa(list(int),list(A),map(int,A,F3),upto(K2,L))) = aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),F3),aa(list(int),set(int),set2(int),upto(K2,L))) ) ).

% interv_sum_list_conv_sum_set_int
tff(fact_7106_size__list__conv__sum__list,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : aa(list(A),nat,size_list(A,F3),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% size_list_conv_sum_list
tff(fact_7107_sum__list__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_js(fun(A,nat),fun(A,nat),F3)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_7108_interv__sum__list__conv__sum__set__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mb: nat,Nb: nat] : groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F3),upt(Mb,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(list(nat),set(nat),set2(nat),upt(Mb,Nb))) ) ).

% interv_sum_list_conv_sum_set_nat
tff(fact_7109_sum__set__upt__conv__sum__list__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(list(nat),set(nat),set2(nat),upt(Mb,Nb))) = groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F3),upt(Mb,Nb))) ) ).

% sum_set_upt_conv_sum_list_nat
tff(fact_7110_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_7111_card__length__sum__list__rec,axiom,
    ! [Mb: nat,N3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Mb)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_afr(nat,fun(nat,fun(list(nat),$o)),Mb),N3))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_afs(nat,fun(nat,fun(list(nat),$o)),Mb),N3)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_aft(nat,fun(nat,fun(list(nat),$o)),Mb),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_7112_card__length__sum__list,axiom,
    ! [Mb: nat,N3: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_afr(nat,fun(nat,fun(list(nat),$o)),Mb),N3))) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),Mb)),one_one(nat))),N3) ).

% card_length_sum_list
tff(fact_7113_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),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_afu(fun(A,nat),fun(list(A),fun(A,nat)),F3),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_7114_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K2: nat,Xs: list(A),Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Xs))
         => ( groups8242544230860333062m_list(A,list_update(A,Xs,K2,Xb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),Xb)),aa(nat,A,nth(A,Xs),K2)) ) ) ) ).

% sum_list_update
tff(fact_7115_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F3: fun(nat,A),Ns: list(nat)] :
          ( ! [X3: nat,Y: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,X3)),aa(nat,A,F3,Y)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F3),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_7116_nth__transpose,axiom,
    ! [A: $tType,I2: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_afv(nat,fun(list(A),A),I2)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afw(nat,fun(list(A),$o),I2)),Xs)) ) ) ).

% nth_transpose
tff(fact_7117_filter__filter,axiom,
    ! [A: $tType,P2: fun(A,$o),Q: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P2),aa(list(A),list(A),filter2(A,Q),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q)),Xs) ).

% filter_filter
tff(fact_7118_filter__True,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X3) )
     => ( aa(list(A),list(A),filter2(A,P2),Xs) = Xs ) ) ).

% filter_True
tff(fact_7119_filter__append,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Ys: list(A)] : aa(list(A),list(A),filter2(A,P2),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),filter2(A,P2),Xs)),aa(list(A),list(A),filter2(A,P2),Ys)) ).

% filter_append
tff(fact_7120_remove1__filter__not,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A)] :
      ( ~ aa(A,$o,P2,Xb)
     => ( remove1(A,Xb,aa(list(A),list(A),filter2(A,P2),Xs)) = aa(list(A),list(A),filter2(A,P2),Xs) ) ) ).

% remove1_filter_not
tff(fact_7121_removeAll__filter__not,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A)] :
      ( ~ aa(A,$o,P2,Xb)
     => ( aa(list(A),list(A),removeAll(A,Xb),aa(list(A),list(A),filter2(A,P2),Xs)) = aa(list(A),list(A),filter2(A,P2),Xs) ) ) ).

% removeAll_filter_not
tff(fact_7122_set__filter,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P2),Xs)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_afy(fun(A,$o),fun(list(A),fun(A,$o)),P2),Xs)) ).

% set_filter
tff(fact_7123_filter__False,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P2,X3) )
     => ( aa(list(A),list(A),filter2(A,P2),Xs) = nil(A) ) ) ).

% filter_False
tff(fact_7124_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P2),aa(list(B),list(A),map(B,A,F3),Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P2),F3)),Xs)) ).

% length_filter_map
tff(fact_7125_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P2: fun(B,$o),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => ( aa(B,$o,P2,Xb)
           => ( aa(list(B),list(B),filter2(B,P2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),aa(list(B),list(B),filter2(B,P2),Xs)) ) ) ) ) ).

% filter_insort
tff(fact_7126_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
        <=> sorted_wrt(B,aTP_Lamp_afz(fun(B,A),fun(B,fun(B,$o)),F3),Xs) ) ) ).

% sorted_map
tff(fact_7127_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),G3: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_aga(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F3),G3),Xs)),Xs))) ) ).

% sorted_map_same
tff(fact_7128_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P2: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P2),Xs))) ) ) ).

% sorted_filter
tff(fact_7129_filter__map,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),filter2(A,P2),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P2),F3)),Xs)) ).

% filter_map
tff(fact_7130_filter__concat,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(list(A))] : aa(list(A),list(A),filter2(A,P),concat(A,Xs)) = concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),filter2(A,P)),Xs)) ).

% filter_concat
tff(fact_7131_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,$o)),F3: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R,aa(list(B),list(A),map(B,A,F3),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_agb(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R),F3),Xs) ) ).

% sorted_wrt_map
tff(fact_7132_distinct__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),P2: fun(B,$o)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
     => distinct(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P2),Xs))) ) ).

% distinct_map_filter
tff(fact_7133_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xb: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ).

% sorted_insort_key
tff(fact_7134_inter__set__filter,axiom,
    ! [A: $tType,A5: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_a(set(A),fun(A,$o),A5)),Xs)) ).

% inter_set_filter
tff(fact_7135_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),remove1(B,Xb,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_7136_filter__shuffles,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Ys: list(A)] : aa(set(list(A)),set(list(A)),image2(list(A),list(A),filter2(A,P2)),shuffles(A,Xs,Ys)) = shuffles(A,aa(list(A),list(A),filter2(A,P2),Xs),aa(list(A),list(A),filter2(A,P2),Ys)) ).

% filter_shuffles
tff(fact_7137_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_7138_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : sorted_wrt(nat,ord_less(nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_7139_filter__insort__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P2: fun(A,$o),Xb: A,F3: fun(A,B),Xs: list(A)] :
          ( ~ aa(A,$o,P2,Xb)
         => ( aa(list(A),list(A),filter2(A,P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Xs)) = aa(list(A),list(A),filter2(A,P2),Xs) ) ) ) ).

% filter_insort_triv
tff(fact_7140_removeAll__filter__not__eq,axiom,
    ! [A: $tType,Xb: A] : removeAll(A,Xb) = filter2(A,aTP_Lamp_agc(A,fun(A,$o),Xb)) ).

% removeAll_filter_not_eq
tff(fact_7141_filter__remove1,axiom,
    ! [A: $tType,Q: fun(A,$o),Xb: A,Xs: list(A)] : aa(list(A),list(A),filter2(A,Q),remove1(A,Xb,Xs)) = remove1(A,Xb,aa(list(A),list(A),filter2(A,Q),Xs)) ).

% filter_remove1
tff(fact_7142_sorted__wrt__filter,axiom,
    ! [A: $tType,F3: fun(A,fun(A,$o)),Xs: list(A),P2: fun(A,$o)] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,aa(list(A),list(A),filter2(A,P2),Xs)) ) ).

% sorted_wrt_filter
tff(fact_7143_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_agd(A,fun(A,$o)),Xs) ).

% sorted_wrt_true
tff(fact_7144_partition__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] : member(list(A),Xs,shuffles(A,aa(list(A),list(A),filter2(A,P2),Xs),aa(list(A),list(A),filter2(A,aTP_Lamp_nj(fun(A,$o),fun(A,$o),P2)),Xs))) ).

% partition_in_shuffles
tff(fact_7145_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_7146_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Nb: nat,Xb: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,Nb,Xb)) ) ).

% sorted_replicate
tff(fact_7147_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_7148_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: 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_mr(A,A)),Xb),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_7149_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G3: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_age(fun(list(A),A),fun(list(A),fun(A,$o)),G3),Xs)),Xs)) ) ).

% sorted_same
tff(fact_7150_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_7151_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_7152_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_7153_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_7154_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Yb: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Yb)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Zs)) ) ) ) ).

% sorted2
tff(fact_7155_filter_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A)] :
      aa(list(A),list(A),filter2(A,P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P2,Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),filter2(A,P2),Xs)),aa(list(A),list(A),filter2(A,P2),Xs)) ).

% filter.simps(2)
tff(fact_7156_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) ) ).

% sorted1
tff(fact_7157_sorted__wrt1,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xb: A] : sorted_wrt(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) ).

% sorted_wrt1
tff(fact_7158_sorted__wrt_Osimps_I1_J,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o))] : sorted_wrt(A,P2,nil(A)) ).

% sorted_wrt.simps(1)
tff(fact_7159_filter_Osimps_I1_J,axiom,
    ! [A: $tType,P2: fun(A,$o)] : aa(list(A),list(A),filter2(A,P2),nil(A)) = nil(A) ).

% filter.simps(1)
tff(fact_7160_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_7161_filter__replicate,axiom,
    ! [A: $tType,P2: fun(A,$o),Nb: nat,Xb: A] :
      aa(list(A),list(A),filter2(A,P2),replicate(A,Nb,Xb)) = $ite(aa(A,$o,P2,Xb),replicate(A,Nb,Xb),nil(A)) ).

% filter_replicate
tff(fact_7162_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_7163_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_7164_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Nb: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,Nb,Xs)) ) ) ).

% sorted_take
tff(fact_7165_sorted__wrt__take,axiom,
    ! [A: $tType,F3: fun(A,fun(A,$o)),Xs: list(A),Nb: nat] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,take(A,Nb,Xs)) ) ).

% sorted_wrt_take
tff(fact_7166_remdups__filter,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : remdups(A,aa(list(A),list(A),filter2(A,P2),Xs)) = aa(list(A),list(A),filter2(A,P2),remdups(A,Xs)) ).

% remdups_filter
tff(fact_7167_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Nb: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,Nb,Xs)) ) ) ).

% sorted_drop
tff(fact_7168_sorted__wrt__drop,axiom,
    ! [A: $tType,F3: fun(A,fun(A,$o)),Xs: list(A),Nb: nat] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,drop(A,Nb,Xs)) ) ).

% sorted_wrt_drop
tff(fact_7169_sum__length__filter__compl,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P2),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aTP_Lamp_nj(fun(A,$o),fun(A,$o),P2)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_7170_replicate__length__filter,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Xb)),Xs)),Xb) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Xb)),Xs) ).

% replicate_length_filter
tff(fact_7171_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_7172_filter__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P2: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,P2,X3)
            <=> aa(A,$o,Q,X3) ) )
       => ( aa(list(A),list(A),filter2(A,P2),Xs) = aa(list(A),list(A),filter2(A,Q),Ys) ) ) ) ).

% filter_cong
tff(fact_7173_filter__id__conv,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P2),Xs) = Xs )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X4) ) ) ).

% filter_id_conv
tff(fact_7174_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,fun(A,$o)),Q: fun(A,fun(A,$o))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y,aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,$o,aa(A,fun(A,$o),P2,X3),Y)
             => aa(A,$o,aa(A,fun(A,$o),Q,X3),Y) ) ) )
     => ( sorted_wrt(A,P2,Xs)
       => sorted_wrt(A,Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_7175_filter__set,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : filter3(A,P2,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P2),Xs)) ).

% filter_set
tff(fact_7176_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_7177_length__filter__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P2: fun(A,$o)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P2,Xb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P2),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_7178_filter__is__subset,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P2),Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_7179_distinct__filter,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),filter2(A,P2),Xs)) ) ).

% distinct_filter
tff(fact_7180_length__filter__le,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P2),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_7181_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => sorted_wrt(A,P2,Xs) ) ).

% sorted_wrt01
tff(fact_7182_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xs: list(A),I2: nat,J: nat] :
      ( sorted_wrt(A,P2,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P2,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_7183_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P2,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P2,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_7184_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_7185_sorted__wrt__append,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( sorted_wrt(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( sorted_wrt(A,P2,Xs)
        & sorted_wrt(A,P2,Ys)
        & ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
           => ! [Xa2: A] :
                ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),P2,X4),Xa2) ) ) ) ) ).

% sorted_wrt_append
tff(fact_7186_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
               => ! [Xa2: A] :
                    ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa2) ) ) ) ) ) ).

% sorted_append
tff(fact_7187_Cons__eq__filterD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P2: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = aa(list(A),list(A),filter2(A,P2),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),Xb),Vs3)) )
          & ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P2,X) )
          & aa(A,$o,P2,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P2),Vs3) ) ) ) ).

% Cons_eq_filterD
tff(fact_7188_filter__eq__ConsD,axiom,
    ! [A: $tType,P2: fun(A,$o),Ys: list(A),Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P2),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),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),Xb),Vs3)) )
          & ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P2,X) )
          & aa(A,$o,P2,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P2),Vs3) ) ) ) ).

% filter_eq_ConsD
tff(fact_7189_Cons__eq__filter__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P2: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = aa(list(A),list(A),filter2(A,P2),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),Xb),Vs2)) )
          & ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P2,X4) )
          & aa(A,$o,P2,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P2),Vs2) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_7190_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Ys: list(A),Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P2),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),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),Xb),Vs2)) )
          & ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P2,X4) )
          & aa(A,$o,P2,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P2),Vs2) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_7191_sorted__wrt_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xb: A,Ys: list(A)] :
      ( sorted_wrt(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))
    <=> ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
           => aa(A,$o,aa(A,fun(A,$o),P2,Xb),X4) )
        & sorted_wrt(A,P2,Ys) ) ) ).

% sorted_wrt.simps(2)
tff(fact_7192_sorted__wrt_Oelims_I3_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( ~ sorted_wrt(A,Xb,Xaa)
     => ~ ! [X3: A,Ys3: list(A)] :
            ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
           => ( ! [Xa3: A] :
                  ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys3))
                 => aa(A,$o,aa(A,fun(A,$o),Xb,X3),Xa3) )
              & sorted_wrt(A,Xb,Ys3) ) ) ) ).

% sorted_wrt.elims(3)
tff(fact_7193_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))
        <=> ( ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X4) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_7194_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))
        <=> ( ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X4) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_7195_empty__filter__conv,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),filter2(A,P2),Xs) )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P2,X4) ) ) ).

% empty_filter_conv
tff(fact_7196_filter__empty__conv,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P2),Xs) = nil(A) )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P2,X4) ) ) ).

% filter_empty_conv
tff(fact_7197_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A2: A,Xs: list(A),F3: fun(A,B)] :
          ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Xs))
           => ( ( aa(list(A),A,hd(A),aa(list(A),list(A),filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_agf(A,fun(fun(A,B),fun(A,$o)),A2),F3)),Xs)) = A2 )
             => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_7198_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Yb: A,Xs: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_agg(fun(A,B),fun(A,fun(A,$o)),F3),Yb)),Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Yb)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_7199_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F3: fun(B,A),P2: fun(B,$o),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P2),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_agh(fun(B,A),fun(fun(B,$o),fun(B,A)),F3),P2)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_7200_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),P2: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),aa(list(A),list(A),filter2(A,P2),Xs)))),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))) ).

% sum_list_filter_le_nat
tff(fact_7201_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S: set(nat)] :
      ( distinct(A,Xs)
     => ( aa(list(A),list(A),filter2(A,aa(set(nat),fun(A,$o),aTP_Lamp_agi(list(A),fun(set(nat),fun(A,$o)),Xs),S)),Xs) = nths(A,Xs,S) ) ) ).

% filter_in_nths
tff(fact_7202_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_7203_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_7204_sorted__wrt_Oelims_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A),Yb: $o] :
      ( ( sorted_wrt(A,Xb,Xaa)
      <=> (Yb) )
     => ( ( ( Xaa = nil(A) )
         => ~ (Yb) )
       => ~ ! [X3: A,Ys3: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
             => ( (Yb)
              <=> ~ ( ! [Xa2: A] :
                        ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys3))
                       => aa(A,$o,aa(A,fun(A,$o),Xb,X3),Xa2) )
                    & sorted_wrt(A,Xb,Ys3) ) ) ) ) ) ).

% sorted_wrt.elims(1)
tff(fact_7205_sorted__wrt_Oelims_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( sorted_wrt(A,Xb,Xaa)
     => ( ( Xaa != nil(A) )
       => ~ ! [X3: A,Ys3: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
             => ~ ( ! [Xa: A] :
                      ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
                     => aa(A,$o,aa(A,fun(A,$o),Xb,X3),Xa) )
                  & sorted_wrt(A,Xb,Ys3) ) ) ) ) ).

% sorted_wrt.elims(2)
tff(fact_7206_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ? [X3: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X3) = A5 )
              & sorted_wrt(A,ord_less_eq(A),X3)
              & distinct(A,X3)
              & ! [Y3: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y3) = A5 )
                    & sorted_wrt(A,ord_less_eq(A),Y3)
                    & distinct(A,Y3) )
                 => ( Y3 = X3 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_7207_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(A),P2: fun(A,$o),F3: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => ( ~ aa(A,$o,P2,X3)
               => ( aa(A,B,F3,X3) = zero_zero(B) ) ) )
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),filter2(A,P2),Xs))) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_7208_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_7209_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Yb: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_agj(A,fun(A,$o),Yb)),Xs)) ).

% set_minus_filter_out
tff(fact_7210_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_agk(list(A),fun(A,$o),Ys)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_7211_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_agl(list(A),fun(A,$o),Ys)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_7212_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_agk(list(A),fun(A,$o),Xs)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_7213_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_agl(list(A),fun(A,$o),Xs)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_7214_filter__eq__nths,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P2),Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_agm(fun(A,$o),fun(list(A),fun(nat,$o)),P2),Xs))) ).

% filter_eq_nths
tff(fact_7215_length__filter__conv__card,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_agm(fun(A,$o),fun(list(A),fun(nat,$o)),P),Xs))) ).

% length_filter_conv_card
tff(fact_7216_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,Xs: list(A)] :
          ( 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_mr(A,A)),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_7217_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_7218_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ~ ! [L3: list(A)] :
                ( sorted_wrt(A,ord_less(A),L3)
               => ( ( aa(list(A),set(A),set2(A),L3) = A5 )
                 => ( aa(list(A),nat,size_size(list(A)),L3) != aa(set(A),nat,finite_card(A),A5) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_7219_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_7220_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_7221_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I2: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(nat,nat,nth(nat,Ns),I2)) ) ) ).

% sorted_wrt_less_idx
tff(fact_7222_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P2),Xs)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P2)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_7223_sorted__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Nb,Xs))) ).

% sorted_enumerate
tff(fact_7224_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A),Ys: list(A)] :
          ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Xs))
           => ( distinct(B,aa(list(A),list(B),map(A,B,F3),Xs))
             => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Ys))
               => ( distinct(B,aa(list(A),list(B),map(A,B,F3),Ys))
                 => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_7225_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A5 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A5) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_7226_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A2) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_mr(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_7227_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aew(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_agn(list(A),$o)),Xss)) ).

% transpose_aux_filter_head
tff(fact_7228_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,F3,Xs) = aa(list(B),list(A),map(B,A,aa(fun(B,option(A)),fun(B,A),comp(option(A),A,B,the2(A)),F3)),aa(list(B),list(B),filter2(B,aTP_Lamp_ago(fun(B,option(A)),fun(B,$o),F3)),Xs)) ).

% map_filter_def
tff(fact_7229_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afw(nat,fun(list(A),$o),I2)),Xs)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I2) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_7230_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_7231_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_7232_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_7233_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_7234_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_7235_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_7236_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_7237_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_7238_rev__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),list(A),rev(A),replicate(A,Nb,Xb)) = replicate(A,Nb,Xb) ).

% rev_replicate
tff(fact_7239_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_7240_inj__on__rev,axiom,
    ! [A: $tType,A5: set(list(A))] : inj_on(list(A),list(A),rev(A),A5) ).

% inj_on_rev
tff(fact_7241_rev__singleton__conv,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) )
    <=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) ) ) ).

% rev_singleton_conv
tff(fact_7242_singleton__rev__conv,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) = aa(list(A),list(A),rev(A),Xs) )
    <=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) = Xs ) ) ).

% singleton_rev_conv
tff(fact_7243_rev__eq__Cons__iff,axiom,
    ! [A: $tType,Xs: list(A),Yb: 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),Yb),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),Yb),nil(A))) ) ) ).

% rev_eq_Cons_iff
tff(fact_7244_rev__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),list(A),rev(A),concat(A,Xs)) = concat(A,aa(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_7245_rev__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rev(B),Xs)) ).

% rev_map
tff(fact_7246_rev__filter,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),filter2(A,P2),Xs)) = aa(list(A),list(A),filter2(A,P2),aa(list(A),list(A),rev(A),Xs)) ).

% rev_filter
tff(fact_7247_sorted__wrt__rev,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P2,aa(list(A),list(A),rev(A),Xs))
    <=> sorted_wrt(A,aTP_Lamp_agp(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P2),Xs) ) ).

% sorted_wrt_rev
tff(fact_7248_sorted__upto,axiom,
    ! [Mb: int,Nb: int] : sorted_wrt(int,ord_less_eq(int),upto(Mb,Nb)) ).

% sorted_upto
tff(fact_7249_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : sorted_wrt(int,ord_less(int),upto(I2,J)) ).

% sorted_wrt_upto
tff(fact_7250_map__of__filter__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K2: B,Z: A,P2: fun(B,fun(A,$o))] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),Z) )
     => ( aa(A,$o,aa(B,fun(A,$o),P2,K2),Z)
       => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),filter2(product_prod(B,A),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P2)),Xs)),K2) = aa(A,option(A),some(A),Z) ) ) ) ).

% map_of_filter_in
tff(fact_7251_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_7252_map__filter__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,option(A))] : map_filter(B,A,F3,nil(B)) = nil(A) ).

% map_filter_simps(2)
tff(fact_7253_rev_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rev(A),nil(A)) = nil(A) ).

% rev.simps(1)
tff(fact_7254_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_7255_rev_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: 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),Xb),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),Xb),nil(A))) ).

% rev.simps(2)
tff(fact_7256_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),Xb: B,Xs: list(B)] : map_filter(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)) = case_option(list(A),A,map_filter(B,A,F3,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_agq(fun(B,option(A)),fun(list(B),fun(A,list(A))),F3),Xs),aa(B,option(A),F3,Xb)) ).

% map_filter_simps(1)
tff(fact_7257_drop__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,Nb,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs)) ).

% drop_rev
tff(fact_7258_rev__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),drop(A,I2,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2),aa(list(A),list(A),rev(A),Xs)) ).

% rev_drop
tff(fact_7259_rev__take,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),take(A,I2,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2),aa(list(A),list(A),rev(A),Xs)) ).

% rev_take
tff(fact_7260_take__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs)) ).

% take_rev
tff(fact_7261_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),map(list(A),nat,size_size(list(A))),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_7262_rev__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rev(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,Nb))) ) ) ).

% rev_nth
tff(fact_7263_rev__update,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),Yb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),rev(A),list_update(A,Xs,K2,Yb)) = 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)),K2)),one_one(nat)),Yb) ) ) ).

% rev_update
tff(fact_7264_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P2: fun(nat,$o),Xs: list(A),Is: list(nat)] : aa(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)),filter2(product_prod(A,nat),aTP_Lamp_agr(fun(nat,$o),fun(product_prod(A,nat),$o),P2)),zip(A,nat,Xs,Is))) = aa(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)),filter2(product_prod(A,nat),aTP_Lamp_ags(fun(nat,$o),fun(product_prod(A,nat),$o),P2)),zip(A,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_7265_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))),aa(nat,A,nth(A,Xs),I4)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_7266_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I2)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_7267_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I4)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_7268_nths__shift__lemma,axiom,
    ! [A: $tType,A5: set(nat),Xs: list(A),I2: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_agt(set(nat),fun(product_prod(A,nat),$o),A5)),zip(A,nat,Xs,upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(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)),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_agu(set(nat),fun(nat,fun(product_prod(A,nat),$o)),A5),I2)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_7269_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat)] : nths(A,Xs,A5) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_agt(set(nat),fun(product_prod(A,nat),$o),A5)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_7270_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),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = $ite(Xs = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_7271_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afw(nat,fun(list(A),$o),I2)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) ) ) ) ).

% transpose_column_length
tff(fact_7272_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P2: fun(B,$o),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P2),Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_agv(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F3),P2),Xs) ).

% map_filter_map_filter
tff(fact_7273_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_afv(nat,fun(list(A),A),I2)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afw(nat,fun(list(A),$o),I2)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I2) ) ) ) ).

% transpose_column
tff(fact_7274_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aex(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_agn(list(A),$o)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_7275_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_agw(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_agn(list(A),$o)),Xs)) ).

% transpose_max_length
tff(fact_7276_tl__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa(list(nat),list(nat),tl(nat),upt(Mb,Nb)) = upt(aa(nat,nat,suc,Mb),Nb) ).

% tl_upt
tff(fact_7277_foldr__append,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),Xs: list(B),Ys: list(B),A2: A] : aa(A,A,foldr(B,A,F3,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),A2) = aa(A,A,foldr(B,A,F3,Xs),aa(A,A,foldr(B,A,F3,Ys),A2)) ).

% foldr_append
tff(fact_7278_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_7279_remdups__adj__Cons__alt,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),tl(A),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)))) = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) ).

% remdups_adj_Cons_alt
tff(fact_7280_foldr__replicate,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Nb: nat,Xb: B] : foldr(B,A,F3,replicate(B,Nb,Xb)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(B,fun(A,A),F3,Xb)) ).

% foldr_replicate
tff(fact_7281_length__concat__rev,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) = aa(list(A),nat,size_size(list(A)),concat(A,Xs)) ).

% length_concat_rev
tff(fact_7282_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_7283_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_7284_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_7285_tl__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),list(A),tl(A),replicate(A,Nb,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xb) ).

% tl_replicate
tff(fact_7286_list_Omap__sel_I2_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F3: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),list(B),tl(B),aa(list(A),list(B),map(A,B,F3),A2)) = aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),tl(A),A2)) ) ) ).

% list.map_sel(2)
tff(fact_7287_map__tl,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),tl(B),Xs)) = aa(list(A),list(A),tl(A),aa(list(B),list(A),map(B,A,F3),Xs)) ).

% map_tl
tff(fact_7288_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_7289_foldr__Cons,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xb: B,Xs: list(B)] : foldr(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(B,fun(A,A),F3,Xb)),foldr(B,A,F3,Xs)) ).

% foldr_Cons
tff(fact_7290_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: fun(B,fun(A,A)),F3: fun(C,B),Xs: list(C),A2: A] : aa(A,A,foldr(B,A,G3,aa(list(C),list(B),map(C,B,F3),Xs)),A2) = aa(A,A,foldr(C,A,aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G3),F3),Xs),A2) ).

% foldr_map
tff(fact_7291_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),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys),aTP_Lamp_agx(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_7292_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_7293_tl__Nil,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ? [X4: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) ) ) ).

% tl_Nil
tff(fact_7294_Nil__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),tl(A),Xs) )
    <=> ( ( Xs = nil(A) )
        | ? [X4: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) ) ) ).

% Nil_tl
tff(fact_7295_list_Osel_I2_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).

% list.sel(2)
tff(fact_7296_foldr__Nil,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A))] : foldr(B,A,F3,nil(B)) = id(A) ).

% foldr_Nil
tff(fact_7297_take__tl,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),take(A,aa(nat,nat,suc,Nb),Xs)) ).

% take_tl
tff(fact_7298_drop__Suc,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,aa(nat,nat,suc,Nb),Xs) = drop(A,Nb,aa(list(A),list(A),tl(A),Xs)) ).

% drop_Suc
tff(fact_7299_tl__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),drop(A,Nb,Xs)) = drop(A,Nb,aa(list(A),list(A),tl(A),Xs)) ).

% tl_drop
tff(fact_7300_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K2: list(B),F3: fun(B,fun(A,A)),G3: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K2 )
       => ( ! [A4: A,X3: B] :
              ( member(B,X3,aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F3,X3),A4) = aa(A,A,aa(B,fun(A,A),G3,X3),A4) ) )
         => ( aa(A,A,foldr(B,A,F3,L),A2) = aa(A,A,foldr(B,A,G3,K2),B2) ) ) ) ) ).

% foldr_cong
tff(fact_7301_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_7302_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A2: list(A),Xb: A] :
      ( ( A2 != nil(A) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),A2)))
       => member(A,Xb,aa(list(A),set(A),set2(A),A2)) ) ) ).

% list.set_sel(2)
tff(fact_7303_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_7304_tl__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_agy(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_7305_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P2: fun(B,$o),Xs: list(B)] : foldr(B,A,F3,aa(list(B),list(B),filter2(B,P2),Xs)) = foldr(B,A,aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_agz(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F3),P2),Xs) ).

% foldr_filter
tff(fact_7306_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_7307_tl__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),take(A,Nb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),aa(list(A),list(A),tl(A),Xs)) ).

% tl_take
tff(fact_7308_list_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      aa(list(B),A,case_list(A,B,F1,F22),List) = $ite(List = nil(B),F1,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ).

% list.case_eq_if
tff(fact_7309_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),A2: A,Xs: list(B)] : foldl(A,B,F3,A2,Xs) = aa(A,A,foldr(B,A,aTP_Lamp_aha(fun(A,fun(B,A)),fun(B,fun(A,A)),F3),aa(list(B),list(B),rev(B),Xs)),A2) ).

% foldl_conv_foldr
tff(fact_7310_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xs: list(B),A2: A] : aa(A,A,foldr(B,A,F3,Xs),A2) = foldl(A,B,aTP_Lamp_ahb(fun(B,fun(A,A)),fun(A,fun(B,A)),F3),A2,aa(list(B),list(B),rev(B),Xs)) ).

% foldr_conv_foldl
tff(fact_7311_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = aa(A,A,foldr(A,A,plus_plus(A),Xs),zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_7312_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)))) ).

% Nitpick.size_list_simp(2)
tff(fact_7313_nth__tl,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,Nb)) ) ) ).

% nth_tl
tff(fact_7314_remdups__adj__append,axiom,
    ! [A: $tType,Xs_1: list(A),Xb: 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),Xb),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),Xb),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),Xb),Xs_2)))) ).

% remdups_adj_append
tff(fact_7315_Cons__in__shuffles__iff,axiom,
    ! [A: $tType,Z: A,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( 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 )
          & 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 )
          & member(list(A),Zs,shuffles(A,Xs,aa(list(A),list(A),tl(A),Ys))) ) ) ) ).

% Cons_in_shuffles_iff
tff(fact_7316_list_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F1: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,P2,aa(list(B),A,case_list(A,B,F1,F22),List))
    <=> ( ( ( List = nil(B) )
         => aa(A,$o,P2,F1) )
        & ( ( List = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List)) )
         => aa(A,$o,P2,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ) ) ).

% list.split_sel
tff(fact_7317_list_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F1: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,P2,aa(list(B),A,case_list(A,B,F1,F22),List))
    <=> ~ ( ( ( List = nil(B) )
            & ~ aa(A,$o,P2,F1) )
          | ( ( List = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List)) )
            & ~ aa(A,$o,P2,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ) ) ).

% list.split_sel_asm
tff(fact_7318_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( Xs != nil(A) )
     => ( take(A,aa(nat,nat,suc,Nb),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),take(A,Nb,aa(list(A),list(A),tl(A),Xs))) ) ) ).

% take_Suc
tff(fact_7319_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_7320_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_ahc(fun(B,A),fun(A,fun(B,fun(A,A))),F3),A2),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_7321_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] :
      aa(list(A),nat,size_list(A,F3),Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F3),aa(list(A),list(A),tl(A),Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_7322_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,foldr(list(A),nat,aTP_Lamp_agw(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_7323_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Yb: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Yb) = $ite(Xs = nil(A),Yb,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Yb)) ) ) ) ).

% foldr_max_sorted
tff(fact_7324_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_ahd(list(B),fun(nat,nat)),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_ahe(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_ahf(list(B),$o)),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_7325_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ~ ! [L3: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L3))
               => ( ( aa(list(B),set(B),set2(B),L3) = A5 )
                 => ( aa(list(B),nat,size_size(list(B)),L3) != aa(set(B),nat,finite_card(B),A5) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_7326_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_agn(list(A),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_7327_takeWhile__idem,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : takeWhile(A,P2,takeWhile(A,P2,Xs)) = takeWhile(A,P2,Xs) ).

% takeWhile_idem
tff(fact_7328_takeWhile__eq__all__conv,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( takeWhile(A,P2,Xs) = Xs )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X4) ) ) ).

% takeWhile_eq_all_conv
tff(fact_7329_takeWhile__append1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P2: fun(A,$o),Ys: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P2,Xb)
       => ( takeWhile(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = takeWhile(A,P2,Xs) ) ) ) ).

% takeWhile_append1
tff(fact_7330_takeWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o),Ys: list(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X3) )
     => ( takeWhile(A,P2,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),takeWhile(A,P2,Ys)) ) ) ).

% takeWhile_append2
tff(fact_7331_takeWhile__replicate,axiom,
    ! [A: $tType,P2: fun(A,$o),Nb: nat,Xb: A] :
      takeWhile(A,P2,replicate(A,Nb,Xb)) = $ite(aa(A,$o,P2,Xb),replicate(A,Nb,Xb),nil(A)) ).

% takeWhile_replicate
tff(fact_7332_takeWhile__map,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F3: fun(B,A),Xs: list(B)] : takeWhile(A,P2,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),takeWhile(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P2),F3),Xs)) ).

% takeWhile_map
tff(fact_7333_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P2,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_7334_zip__takeWhile__snd,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),P2: fun(B,$o),Ys: list(B)] : zip(A,B,Xs,takeWhile(B,P2,Ys)) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),$o),comp(B,$o,product_prod(A,B),P2),product_snd(A,B)),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_snd
tff(fact_7335_zip__takeWhile__fst,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),Xs: list(A),Ys: list(B)] : zip(A,B,takeWhile(A,P2,Xs),Ys) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),$o),comp(A,$o,product_prod(A,B),P2),product_fst(A,B)),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_fst
tff(fact_7336_takeWhile__eq__Nil__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( takeWhile(A,P2,Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ~ aa(A,$o,P2,aa(list(A),A,hd(A),Xs)) ) ) ).

% takeWhile_eq_Nil_iff
tff(fact_7337_length__takeWhile__le,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_takeWhile_le
tff(fact_7338_distinct__takeWhile,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,takeWhile(A,P2,Xs)) ) ).

% distinct_takeWhile
tff(fact_7339_takeWhile__cong,axiom,
    ! [A: $tType,L: list(A),K2: list(A),P2: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K2 )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P2,X3)
            <=> aa(A,$o,Q,X3) ) )
       => ( takeWhile(A,P2,L) = takeWhile(A,Q,K2) ) ) ) ).

% takeWhile_cong
tff(fact_7340_set__takeWhileD,axiom,
    ! [A: $tType,Xb: A,P2: fun(A,$o),Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),takeWhile(A,P2,Xs)))
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
        & aa(A,$o,P2,Xb) ) ) ).

% set_takeWhileD
tff(fact_7341_takeWhile__eq__take,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : takeWhile(A,P2,Xs) = take(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)),Xs) ).

% takeWhile_eq_take
tff(fact_7342_takeWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P2: fun(A,$o)] : takeWhile(A,P2,nil(A)) = nil(A) ).

% takeWhile.simps(1)
tff(fact_7343_takeWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A)] :
      takeWhile(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P2,Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),takeWhile(A,P2,Xs)),nil(A)) ).

% takeWhile.simps(2)
tff(fact_7344_folding__insort__key_Oinj__on,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => inj_on(B,A,F3,S2) ) ).

% folding_insort_key.inj_on
tff(fact_7345_takeWhile__tail,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A),L: list(A)] :
      ( ~ aa(A,$o,P2,Xb)
     => ( takeWhile(A,P2,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),Xb),L))) = takeWhile(A,P2,Xs) ) ) ).

% takeWhile_tail
tff(fact_7346_folding__insort__key_Odistinct__if__distinct__map,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
       => distinct(B,Xs) ) ) ).

% folding_insort_key.distinct_if_distinct_map
tff(fact_7347_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P2: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))
     => ( aa(nat,A,nth(A,takeWhile(A,P2,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_7348_nth__length__takeWhile,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))
     => ~ aa(A,$o,P2,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))) ) ).

% nth_length_takeWhile
tff(fact_7349_takeWhile__append,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Ys: list(A)] :
      takeWhile(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(
        ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X4) ),
        aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P2,Ys)),
        takeWhile(A,P2,Xs) ) ).

% takeWhile_append
tff(fact_7350_concat__conv__foldr,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,Xss) = aa(list(A),list(A),foldr(list(A),list(A),append(A),Xss),nil(A)) ).

% concat_conv_foldr
tff(fact_7351_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P2: fun(A,$o),Xs: list(A)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J)
         => aa(A,$o,P2,aa(nat,A,nth(A,Xs),I3)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_7352_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P2: fun(A,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,P2,aa(nat,A,nth(A,Xs),I3)) ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
         => ~ aa(A,$o,P2,aa(nat,A,nth(A,Xs),Nb)) )
       => ( takeWhile(A,P2,Xs) = take(A,Nb,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_7353_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_mr(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_7354_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Ta: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F3),Xs)))
         => ( aa(list(B),list(B),filter2(B,aa(A,fun(B,$o),aTP_Lamp_ahg(fun(B,A),fun(A,fun(B,$o)),F3),Ta)),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_ahg(fun(B,A),fun(A,fun(B,$o)),F3),Ta),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_7355_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L))
              & ( aa(list(B),set(B),set2(B),L) = A5 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A5) ) )
          <=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_7356_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xb: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),A5)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),bot_bot(set(B))))) = remove1(B,Xb,aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_7357_linorder_Osorted__key__list__of__set_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : sorted8670434370408473282of_set(B,A,Less_eq) = sorted8670434370408473282of_set(B,A,Less_eq) ).

% linorder.sorted_key_list_of_set.cong
tff(fact_7358_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B),B5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),S2)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),B5) )
           => ( aa(set(B),$o,finite_finite2(B),A5)
             => ( aa(set(B),$o,finite_finite2(B),B5)
               => ( A5 = B5 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_7359_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_7360_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( 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),F3),A5)) = A5 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_7361_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),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),F3),A5)) = aa(set(B),nat,finite_card(B),A5) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_7362_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => distinct(A,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_7363_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_7364_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_7365_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5) = nil(B) )
          <=> ( A5 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7366_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S2)
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),Xs))
         => ( distinct(B,Xs)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_7367_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xb: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),A5)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),A5)) = 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),F3),Xb),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_7368_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xb: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),A5)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( ~ member(B,Xb,A5)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),A5)) = 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),F3),Xb),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_7369_linorder_Oinsort__key_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : insort_key(B,A,Less_eq) = insort_key(B,A,Less_eq) ).

% linorder.insort_key.cong
tff(fact_7370_folding__insort__key_Oinsort__key__commute,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xb: B,Yb: B] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( member(B,Xb,S2)
       => ( member(B,Yb,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),F3),Yb)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Xb)) = 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),F3),Xb)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Yb)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_7371_extract__def,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : extract(A,P2,Xs) = aa(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_ahh(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P2),Xs)),dropWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P2),Xs)) ).

% extract_def
tff(fact_7372_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
                & aa(A,$o,P2,X) )
           => ( find(A,P2,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahi(list(A),fun(fun(A,$o),fun(A,$o)),Xs),P2)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_7373_dropWhile__idem,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : dropWhile(A,P2,dropWhile(A,P2,Xs)) = dropWhile(A,P2,Xs) ).

% dropWhile_idem
tff(fact_7374_dropWhile__eq__Nil__conv,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P2,Xs) = nil(A) )
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X4) ) ) ).

% dropWhile_eq_Nil_conv
tff(fact_7375_dropWhile__append1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P2: fun(A,$o),Ys: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P2,Xb)
       => ( dropWhile(A,P2,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),dropWhile(A,P2,Xs)),Ys) ) ) ) ).

% dropWhile_append1
tff(fact_7376_dropWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o),Ys: list(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X3) )
     => ( dropWhile(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = dropWhile(A,P2,Ys) ) ) ).

% dropWhile_append2
tff(fact_7377_dropWhile__replicate,axiom,
    ! [A: $tType,P2: fun(A,$o),Nb: nat,Xb: A] :
      dropWhile(A,P2,replicate(A,Nb,Xb)) = $ite(aa(A,$o,P2,Xb),nil(A),replicate(A,Nb,Xb)) ).

% dropWhile_replicate
tff(fact_7378_takeWhile__dropWhile__id,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),takeWhile(A,P2,Xs)),dropWhile(A,P2,Xs)) = Xs ).

% takeWhile_dropWhile_id
tff(fact_7379_dropWhile__append3,axiom,
    ! [A: $tType,P2: fun(A,$o),Yb: A,Xs: list(A),Ys: list(A)] :
      ( ~ aa(A,$o,P2,Yb)
     => ( dropWhile(A,P2,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),Yb),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) ) ) ).

% dropWhile_append3
tff(fact_7380_remdups__adj__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),remdups_adj(A,dropWhile(A,aTP_Lamp_cc(A,fun(A,$o),Xb),Xs))) ).

% remdups_adj_Cons'
tff(fact_7381_dropWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A)] :
      dropWhile(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P2,Xb),dropWhile(A,P2,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) ).

% dropWhile.simps(2)
tff(fact_7382_dropWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P2: fun(A,$o)] : dropWhile(A,P2,nil(A)) = nil(A) ).

% dropWhile.simps(1)
tff(fact_7383_find__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P2: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,P2,X3)
            <=> aa(A,$o,Q,X3) ) )
       => ( find(A,P2,Xs) = find(A,Q,Ys) ) ) ) ).

% find_cong
tff(fact_7384_dropWhile__cong,axiom,
    ! [A: $tType,L: list(A),K2: list(A),P2: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K2 )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P2,X3)
            <=> aa(A,$o,Q,X3) ) )
       => ( dropWhile(A,P2,L) = dropWhile(A,Q,K2) ) ) ) ).

% dropWhile_cong
tff(fact_7385_set__dropWhileD,axiom,
    ! [A: $tType,Xb: A,P2: fun(A,$o),Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),dropWhile(A,P2,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_dropWhileD
tff(fact_7386_distinct__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,dropWhile(A,P2,Xs)) ) ).

% distinct_dropWhile
tff(fact_7387_length__dropWhile__le,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_dropWhile_le
tff(fact_7388_hd__dropWhile,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P2,Xs) != nil(A) )
     => ~ aa(A,$o,P2,aa(list(A),A,hd(A),dropWhile(A,P2,Xs))) ) ).

% hd_dropWhile
tff(fact_7389_dropWhile__eq__self__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P2,Xs) = Xs )
    <=> ( ( Xs = nil(A) )
        | ~ aa(A,$o,P2,aa(list(A),A,hd(A),Xs)) ) ) ).

% dropWhile_eq_self_iff
tff(fact_7390_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P2,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_7391_dropWhile__map,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F3: fun(B,A),Xs: list(B)] : dropWhile(A,P2,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),dropWhile(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P2),F3),Xs)) ).

% dropWhile_map
tff(fact_7392_find__dropWhile,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : find(A,P2,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_ahj(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P2),Xs)) ).

% find_dropWhile
tff(fact_7393_find_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A)] :
      find(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P2,Xb),aa(A,option(A),some(A),Xb),find(A,P2,Xs)) ).

% find.simps(2)
tff(fact_7394_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,$o)] : find(A,Uu,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_7395_find__None__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( find(A,P2,Xs) = none(A) )
    <=> ~ ? [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P2,X4) ) ) ).

% find_None_iff
tff(fact_7396_find__None__iff2,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ( none(A) = find(A,P2,Xs) )
    <=> ~ ? [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P2,X4) ) ) ).

% find_None_iff2
tff(fact_7397_dropWhile__eq__Cons__conv,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Yb: A,Ys: list(A)] :
      ( ( dropWhile(A,P2,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),takeWhile(A,P2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) )
        & ~ aa(A,$o,P2,Yb) ) ) ).

% dropWhile_eq_Cons_conv
tff(fact_7398_takeWhile__eq__filter,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),dropWhile(A,P2,Xs)))
         => ~ aa(A,$o,P2,X3) )
     => ( takeWhile(A,P2,Xs) = aa(list(A),list(A),filter2(A,P2),Xs) ) ) ).

% takeWhile_eq_filter
tff(fact_7399_dropWhile__eq__drop,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : dropWhile(A,P2,Xs) = drop(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)),Xs) ).

% dropWhile_eq_drop
tff(fact_7400_dropWhile__append,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Ys: list(A)] :
      dropWhile(A,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(
        ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X4) ),
        dropWhile(A,P2,Ys),
        aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P2,Xs)),Ys) ) ).

% dropWhile_append
tff(fact_7401_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Yb: A,Ys: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),nil(A))))),remdups_adj(A,dropWhile(A,aTP_Lamp_cc(A,fun(A,$o),Yb),Ys))) ).

% remdups_adj_append_dropWhile
tff(fact_7402_tl__remdups__adj,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( ( Ys != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys)) = remdups_adj(A,dropWhile(A,aTP_Lamp_ahk(list(A),fun(A,$o),Ys),aa(list(A),list(A),tl(A),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_7403_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P2: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),dropWhile(A,P2,Xs)))
     => ( aa(nat,A,nth(A,dropWhile(A,P2,Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_7404_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( dropWhile(A,aTP_Lamp_agj(A,fun(A,$o),Xb),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),rev(A),takeWhile(A,aTP_Lamp_agj(A,fun(A,$o),Xb),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_7405_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( takeWhile(A,aTP_Lamp_agj(A,fun(A,$o),Xb),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),dropWhile(A,aTP_Lamp_agj(A,fun(A,$o),Xb),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_7406_find__Some__iff2,axiom,
    ! [A: $tType,Xb: A,P2: fun(A,$o),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),Xb) = find(A,P2,Xs) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P2,aa(nat,A,nth(A,Xs),I4))
          & ( Xb = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P2,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_7407_find__Some__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Xb: A] :
      ( ( find(A,P2,Xs) = aa(A,option(A),some(A),Xb) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P2,aa(nat,A,nth(A,Xs),I4))
          & ( Xb = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P2,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_7408_partition__filter__conv,axiom,
    ! [A: $tType,F3: fun(A,$o),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F3),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),filter2(A,F3),Xs)),aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),F3)),Xs)) ).

% partition_filter_conv
tff(fact_7409_lenlex__append2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys: list(A)] :
      ( irrefl(A,R)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(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))
      <=> 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_7410_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)
     => ( 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))
      <=> 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_7411_partition__filter1,axiom,
    ! [A: $tType,P2: fun(A,$o),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,P2),Xs)) = aa(list(A),list(A),filter2(A,P2),Xs) ).

% partition_filter1
tff(fact_7412_irreflI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A4: A] : ~ 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_7413_irrefl__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [A7: A] : ~ 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_7414_lexord__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
     => irrefl(list(A),lexord(A,R)) ) ).

% lexord_irrefl
tff(fact_7415_irrefl__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
     => irrefl(list(A),lex(A,R2)) ) ).

% irrefl_lex
tff(fact_7416_partition_Osimps_I1_J,axiom,
    ! [A: $tType,P2: fun(A,$o)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P2),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_7417_partition__P,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P2),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) )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Yes))
           => aa(A,$o,P2,X) )
        & ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),No4))
           => ~ aa(A,$o,P2,X) ) ) ) ).

% partition_P
tff(fact_7418_lexl__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: list(A)] :
      ( irrefl(A,R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Xb),lex(A,R2)) ) ).

% lexl_not_refl
tff(fact_7419_partition_Osimps_I2_J,axiom,
    ! [A: $tType,P2: fun(A,$o),Xb: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),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_ahl(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P2),Xb)),aa(list(A),product_prod(list(A),list(A)),partition(A,P2),Xs)) ).

% partition.simps(2)
tff(fact_7420_partition__filter2,axiom,
    ! [A: $tType,P2: fun(A,$o),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,P2),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P2)),Xs) ).

% partition_filter2
tff(fact_7421_partition__set,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P2),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_7422_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Yb: A] :
          ( ( min_list(A,Xb) = Yb )
         => ( ! [X3: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( Yb != aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_aev(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) ) )
           => ~ ( ( Xb = nil(A) )
               => ( Yb != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_7423_lists__length__Suc__eq,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] : aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ahm(set(A),fun(nat,fun(list(A),$o)),A5),Nb)) = 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_aed(list(A),fun(A,list(A))))),product_Sigma(list(A),A,aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_au(set(A),fun(nat,fun(list(A),$o)),A5),Nb)),aTP_Lamp_ahn(set(A),fun(list(A),set(A)),A5))) ).

% lists_length_Suc_eq
tff(fact_7424_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A5: set(A),B5: fun(A,set(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),product_Sigma(A,B,A5,B5))
    <=> ( member(A,A2,A5)
        & member(B,B2,aa(A,set(B),B5,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_7425_SigmaI,axiom,
    ! [B: $tType,A: $tType,A2: A,A5: set(A),B2: B,B5: fun(A,set(B))] :
      ( member(A,A2,A5)
     => ( member(B,B2,aa(A,set(B),B5,A2))
       => 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,A5,B5)) ) ) ).

% SigmaI
tff(fact_7426_Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),Q: fun(B,$o)] : aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_aho(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),P2),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P2),aTP_Lamp_ahp(fun(B,$o),fun(A,set(B)),Q)) ).

% Collect_case_prod
tff(fact_7427_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B5: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B5) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_7428_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_ahq(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A5),aTP_Lamp_ahq(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_7429_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A5: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahr(set(B),fun(A,set(B)),A5))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahs(set(B),fun(A,set(B)),A5)) ).

% Compl_Times_UNIV1
tff(fact_7430_Times__empty,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( ( product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( B5 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_7431_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : product_Sigma(A,B,A5,aTP_Lamp_aht(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_7432_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [A4: A] :
            ( member(A,A4,A5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,A4)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,B5)) ) ) ).

% finite_SigmaI
tff(fact_7433_UNIV__Times__UNIV,axiom,
    ! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahq(A,set(B))) = top_top(set(product_prod(A,B))) ).

% UNIV_Times_UNIV
tff(fact_7434_fst__image__times,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
      aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5))) = $ite(B5 = bot_bot(set(B)),bot_bot(set(A)),A5) ).

% fst_image_times
tff(fact_7435_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A5: set(B),B5: set(A)] :
      aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A5,aTP_Lamp_ms(set(A),fun(B,set(A)),B5))) = $ite(A5 = bot_bot(set(B)),bot_bot(set(A)),B5) ).

% snd_image_times
tff(fact_7436_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_ahu(list(B),fun(A,set(B)),Ys)) ).

% set_product
tff(fact_7437_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A2: A,A5: set(A),B2: B,B5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5),aa(set(B),fun(A,set(B)),aTP_Lamp_ahv(B,fun(set(B),fun(A,set(B))),B2),B5)) = 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))),insert(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,A5,aa(set(B),fun(A,set(B)),aTP_Lamp_ahv(B,fun(set(B),fun(A,set(B))),B2),B5))),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5),aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)))) ).

% insert_Times_insert
tff(fact_7438_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X3: A] :
            ( member(A,X3,A5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,X3)) )
       => ( aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A5,B5)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_nf(fun(A,set(B)),fun(A,nat),B5)),A5) ) ) ) ).

% card_SigmaI
tff(fact_7439_inj__on__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C),A5: set(A)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),product_Sigma(A,B,A5,aTP_Lamp_ahq(A,set(B))))
    <=> inj_on(A,C,F3,A5) ) ).

% inj_on_apfst
tff(fact_7440_inj__on__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C),A5: set(B)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),product_apsnd(B,C,A,F3),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahr(set(B),fun(A,set(B)),A5)))
    <=> inj_on(B,C,F3,A5) ) ).

% inj_on_apsnd
tff(fact_7441_Collect__case__prod__Sigma,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),Q: fun(A,fun(B,$o))] : aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ahw(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),P2),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P2),aTP_Lamp_ahx(fun(A,fun(B,$o)),fun(A,set(B)),Q)) ).

% Collect_case_prod_Sigma
tff(fact_7442_Times__Int__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5),aTP_Lamp_ahr(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_ahr(set(B),fun(A,set(B)),C5))) ).

% Times_Int_distrib1
tff(fact_7443_Sigma__Int__distrib2,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I6,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ahy(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I6,A5)),product_Sigma(A,B,I6,B5)) ).

% Sigma_Int_distrib2
tff(fact_7444_Times__Int__Times,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),C5: set(A),D5: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5))),product_Sigma(A,B,C5,aTP_Lamp_ahr(set(B),fun(A,set(B)),D5))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C5),aa(set(B),fun(A,set(B)),aTP_Lamp_ahz(set(B),fun(set(B),fun(A,set(B))),B5),D5)) ).

% Times_Int_Times
tff(fact_7445_Sigma__Int__distrib1,axiom,
    ! [B: $tType,A: $tType,I6: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),I6),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I6,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Int_distrib1
tff(fact_7446_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I6: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I6,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Diff_distrib1
tff(fact_7447_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5),aTP_Lamp_ahr(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_ahr(set(B),fun(A,set(B)),C5))) ).

% Times_Diff_distrib1
tff(fact_7448_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I6,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_aia(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I6,A5)),product_Sigma(A,B,I6,B5)) ).

% Sigma_Diff_distrib2
tff(fact_7449_Sigma__cong,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(A),C5: fun(A,set(B)),D5: fun(A,set(B))] :
      ( ( A5 = B5 )
     => ( ! [X3: A] :
            ( member(A,X3,B5)
           => ( aa(A,set(B),C5,X3) = aa(A,set(B),D5,X3) ) )
       => ( product_Sigma(A,B,A5,C5) = product_Sigma(A,B,B5,D5) ) ) ) ).

% Sigma_cong
tff(fact_7450_Times__eq__cancel2,axiom,
    ! [A: $tType,B: $tType,Xb: A,C5: set(A),A5: set(B),B5: set(B)] :
      ( member(A,Xb,C5)
     => ( ( product_Sigma(B,A,A5,aTP_Lamp_ms(set(A),fun(B,set(A)),C5)) = product_Sigma(B,A,B5,aTP_Lamp_ms(set(A),fun(B,set(A)),C5)) )
      <=> ( A5 = B5 ) ) ) ).

% Times_eq_cancel2
tff(fact_7451_finite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5))) ) ) ).

% finite_cartesian_product
tff(fact_7452_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5),aTP_Lamp_ahr(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_ahr(set(B),fun(A,set(B)),C5))) ).

% Times_Un_distrib1
tff(fact_7453_Sigma__Un__distrib2,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I6,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_aib(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I6,A5)),product_Sigma(A,B,I6,B5)) ).

% Sigma_Un_distrib2
tff(fact_7454_member__product,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),A5: set(A),B5: set(B)] :
      ( member(product_prod(A,B),Xb,product_product(A,B,A5,B5))
    <=> member(product_prod(A,B),Xb,product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5))) ) ).

% member_product
tff(fact_7455_Product__Type_Oproduct__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] : product_product(A,B,A5,B5) = product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)) ).

% Product_Type.product_def
tff(fact_7456_Sigma__Un__distrib1,axiom,
    ! [B: $tType,A: $tType,I6: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I6),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I6,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Un_distrib1
tff(fact_7457_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,Xb: A,C5: set(A),A5: set(B),B5: set(B)] :
      ( member(A,Xb,C5)
     => ( aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A5,aTP_Lamp_ms(set(A),fun(B,set(A)),C5))),product_Sigma(B,A,B5,aTP_Lamp_ms(set(A),fun(B,set(A)),C5)))
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),B5) ) ) ).

% Times_subset_cancel2
tff(fact_7458_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C5: set(A),B5: fun(A,set(B)),D5: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
     => ( ! [X3: A] :
            ( member(A,X3,A5)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B5,X3)),aa(A,set(B),D5,X3)) )
       => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A5,B5)),product_Sigma(A,B,C5,D5)) ) ) ).

% Sigma_mono
tff(fact_7459_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I6: set(A),X5: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I6,X5) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X4: A] :
          ( member(A,X4,I6)
         => ( aa(A,set(B),X5,X4) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_7460_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),C5: set(A),D5: set(B)] :
      ( ( product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)) = product_Sigma(A,B,C5,aTP_Lamp_ahr(set(B),fun(A,set(B)),D5)) )
    <=> ( ( ( A5 = C5 )
          & ( B5 = D5 ) )
        | ( ( ( A5 = bot_bot(set(A)) )
            | ( B5 = bot_bot(set(B)) ) )
          & ( ( C5 = bot_bot(set(A)) )
            | ( D5 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_7461_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A5: set(A),B5: fun(A,set(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),product_Sigma(A,B,A5,B5))
     => ~ ( member(A,A2,A5)
         => ~ member(B,B2,aa(A,set(B),B5,A2)) ) ) ).

% SigmaE2
tff(fact_7462_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A5: set(A),B5: fun(A,set(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),product_Sigma(A,B,A5,B5))
     => member(B,B2,aa(A,set(B),B5,A2)) ) ).

% SigmaD2
tff(fact_7463_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A5: set(A),B5: fun(A,set(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),product_Sigma(A,B,A5,B5))
     => member(A,A2,A5) ) ).

% SigmaD1
tff(fact_7464_SigmaE,axiom,
    ! [A: $tType,B: $tType,C3: product_prod(A,B),A5: set(A),B5: fun(A,set(B))] :
      ( member(product_prod(A,B),C3,product_Sigma(A,B,A5,B5))
     => ~ ! [X3: A] :
            ( member(A,X3,A5)
           => ! [Y: B] :
                ( member(B,Y,aa(A,set(B),B5,X3))
               => ( C3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) ) ) ) ) ).

% SigmaE
tff(fact_7465_Sigma__Union,axiom,
    ! [B: $tType,A: $tType,X5: set(set(A)),B5: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X5),B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(set(A)),set(set(product_prod(A,B))),image2(set(A),set(product_prod(A,B)),aTP_Lamp_aic(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B5)),X5)) ).

% Sigma_Union
tff(fact_7466_mem__Times__iff,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),A5: set(A),B5: set(B)] :
      ( member(product_prod(A,B),Xb,product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)))
    <=> ( member(A,aa(product_prod(A,B),A,product_fst(A,B),Xb),A5)
        & member(B,aa(product_prod(A,B),B,product_snd(A,B),Xb),B5) ) ) ).

% mem_Times_iff
tff(fact_7467_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),C5: set(B),B5: set(A),D5: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_ahr(set(B),fun(A,set(B)),D5)))
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( C5 = bot_bot(set(B)) )
        | ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C5),D5) ) ) ) ).

% times_subset_iff
tff(fact_7468_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = case_option(A,A,undefined(A),aTP_Lamp_adm(A,A),Option) ).

% option.the_def
tff(fact_7469_hd__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,case_list(A,A,undefined(A),aTP_Lamp_aid(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_7470_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),A5: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_aie(set(A),fun(A,set(A)),A5)))
       => ( ( A2 = B2 )
          | member(A,A2,A5) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_7471_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)))
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( B5 = bot_bot(set(B)) )
        | ( aa(set(A),$o,finite_finite2(A),A5)
          & aa(set(B),$o,finite_finite2(B),B5) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_7472_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)))
     => ( ( A5 != bot_bot(set(A)) )
       => aa(set(B),$o,finite_finite2(B),B5) ) ) ).

% finite_cartesian_productD2
tff(fact_7473_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)))
     => ( ( B5 != bot_bot(set(B)) )
       => aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% finite_cartesian_productD1
tff(fact_7474_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_aif(set(A),fun(fun(A,set(B)),fun(A,$o)),A5),B5)))
     => ( ! [A4: A] :
            ( member(A,A4,A5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,A4)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,B5)) ) ) ).

% finite_SigmaI2
tff(fact_7475_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A5,B5)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_aif(set(A),fun(fun(A,set(B)),fun(A,$o)),A5),B5)) ).

% fst_image_Sigma
tff(fact_7476_UN__Times__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,E5: fun(C,set(A)),F4: fun(D,set(B)),A5: set(C),B5: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(set(product_prod(A,B))),image2(product_prod(C,D),set(product_prod(A,B)),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_aih(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E5),F4))),product_Sigma(C,D,A5,aTP_Lamp_aii(set(D),fun(C,set(D)),B5)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),E5),A5)),aa(set(D),fun(A,set(B)),aTP_Lamp_aij(fun(D,set(B)),fun(set(D),fun(A,set(B))),F4),B5)) ).

% UN_Times_distrib
tff(fact_7477_swap__product,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B5: 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_ym(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A5,aTP_Lamp_ms(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_ahr(set(B),fun(A,set(B)),A5)) ).

% swap_product
tff(fact_7478_map__prod__surj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(B,A),A5: set(B),A10: set(A),G3: fun(D,C),B5: set(D),B12: set(C)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),A5) = A10 )
     => ( ( aa(set(D),set(C),image2(D,C,G3),B5) = B12 )
       => ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F3,G3)),product_Sigma(B,D,A5,aTP_Lamp_aik(set(D),fun(B,set(D)),B5))) = product_Sigma(A,C,A10,aTP_Lamp_ail(set(C),fun(A,set(C)),B12)) ) ) ) ).

% map_prod_surj_on
tff(fact_7479_map__prod__inj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F3: fun(A,B),A5: set(A),G3: fun(C,D),B5: set(C)] :
      ( inj_on(A,B,F3,A5)
     => ( inj_on(C,D,G3,B5)
       => inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F3,G3),product_Sigma(A,C,A5,aTP_Lamp_ail(set(C),fun(A,set(C)),B5))) ) ) ).

% map_prod_inj_on
tff(fact_7480_snd__image__Sigma,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B5: fun(B,set(A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A5,B5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A5)) ).

% snd_image_Sigma
tff(fact_7481_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A5),aTP_Lamp_aim(set(product_prod(A,B)),fun(A,set(B)),A5))) ).

% subset_fst_snd
tff(fact_7482_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,Xb: A,A5: 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)),insert(A),Xb),bot_bot(set(A))),aTP_Lamp_ahr(set(B),fun(A,set(B)),A5))) = aa(set(B),nat,finite_card(B),A5) ).

% card_cartesian_product_singleton
tff(fact_7483_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A5: set(C),B5: 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_adq(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3))),product_Sigma(C,D,A5,aTP_Lamp_aii(set(D),fun(C,set(D)),B5))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F3),A5),aa(set(D),fun(A,set(B)),aTP_Lamp_ain(fun(D,B),fun(set(D),fun(A,set(B))),G3),B5)) ).

% image_paired_Times
tff(fact_7484_Sigma__interval__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [A5: set(A),V: fun(A,B),W2: B] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_aio(fun(A,B),fun(A,set(B)),V))),product_Sigma(A,B,A5,aa(B,fun(A,set(B)),aTP_Lamp_aip(fun(A,B),fun(B,fun(A,set(B))),V),W2))) = bot_bot(set(product_prod(A,B))) ) ).

% Sigma_interval_disjoint
tff(fact_7485_sum_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A5: set(A),B5: fun(A,set(B)),G3: fun(A,fun(B,C))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,X3)) )
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_aiq(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B5),G3)),A5) = aa(set(product_prod(A,B)),C,aa(fun(product_prod(A,B),C),fun(set(product_prod(A,B)),C),groups7311177749621191930dd_sum(product_prod(A,B),C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G3)),product_Sigma(A,B,A5,B5)) ) ) ) ) ).

% sum.Sigma
tff(fact_7486_prod_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A5: set(A),B5: fun(A,set(B)),G3: fun(A,fun(B,C))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,X3)) )
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_air(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B5),G3)),A5) = aa(set(product_prod(A,B)),C,aa(fun(product_prod(A,B),C),fun(set(product_prod(A,B)),C),groups7121269368397514597t_prod(product_prod(A,B),C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G3)),product_Sigma(A,B,A5,B5)) ) ) ) ) ).

% prod.Sigma
tff(fact_7487_Func__empty,axiom,
    ! [B: $tType,A: $tType,B5: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B5) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert(fun(A,B)),aTP_Lamp_ais(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_7488_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] : product_Sigma(A,B,A5,B5) = 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_aiu(fun(A,set(B)),fun(A,set(product_prod(A,B))),B5)),A5)) ).

% Sigma_def
tff(fact_7489_product__fold,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_aiw(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B5),bot_bot(set(product_prod(A,B))),A5) ) ) ) ).

% product_fold
tff(fact_7490_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: fun(A,B),Xaa: list(A),Yb: A] :
          ( ( arg_min_list(A,B,Xb,Xaa) = Yb )
         => ( ! [X3: A] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
               => ( Yb != X3 ) )
           => ( ! [X3: A,Y: A,Zs2: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2)) )
                 => ( Yb != $let(
                        m2: A,
                        m2:= arg_min_list(A,B,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2)),
                        $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Xb,X3)),aa(A,B,Xb,m2)),X3,m2) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ( Yb != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_7491_uniformly__continuous__on__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S: set(A),F3: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S,F3)
        <=> 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_aix(fun(A,B),fun(A,fun(A,product_prod(B,B))),F3)),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_aiy(set(A),fun(A,set(A)),S))))) ) ) ).

% uniformly_continuous_on_uniformity
tff(fact_7492_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Yb: A] :
          ( ( min_list(A,Xb) = Yb )
         => ( accp(list(A),min_list_rel(A),Xb)
           => ( ! [X3: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ( ( Yb = aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_aev(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) )
                   => ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)) ) )
             => ~ ( ( Xb = nil(A) )
                 => ( ( Yb = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_7493_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( ~ aa(set(B),$o,finite_finite2(B),B5)
       => ~ aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_ahr(set(B),fun(A,set(B)),B5))) ) ) ).

% infinite_cartesian_product
tff(fact_7494_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_fr(nat,fun(nat,fun(nat,$o)),Mb))) = product_Sigma(nat,nat,aa(nat,set(nat),set_ord_atMost(nat),Mb),aTP_Lamp_aiz(nat,fun(nat,set(nat)),Mb)) ).

% pairs_le_eq_Sigma
tff(fact_7495_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R2: set(product_prod(A,A)),As3: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R2,As3)
        <=> ! [I4: A,J3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I4),J3),R2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As3,I4)),aa(A,B,As3,J3)) ) ) ) ).

% relChain_def
tff(fact_7496_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat))) ).

% natLess_def
tff(fact_7497_swap__comp__swap,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B)),comp(product_prod(B,A),product_prod(A,B),product_prod(A,B),product_swap(B,A)),product_swap(A,B)) = id(product_prod(A,B)) ).

% swap_comp_swap
tff(fact_7498_rotate__drop__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Nb),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)),Xs)),take(A,modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)),Xs)) ).

% rotate_drop_take
tff(fact_7499_swap__swap,axiom,
    ! [B: $tType,A: $tType,P: product_prod(A,B)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P)) = P ).

% swap_swap
tff(fact_7500_rotate__is__Nil__conv,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( aa(list(A),list(A),rotate(A,Nb),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rotate_is_Nil_conv
tff(fact_7501_set__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rotate
tff(fact_7502_length__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate
tff(fact_7503_swap__simp,axiom,
    ! [A: $tType,B: $tType,Xb: B,Yb: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Yb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Yb),Xb) ).

% swap_simp
tff(fact_7504_distinct__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rotate(A,Nb),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_rotate
tff(fact_7505_case__swap,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,fun(B,A)),P: product_prod(C,B)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aTP_Lamp_aja(fun(C,fun(B,A)),fun(B,fun(C,A)),F3)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F3),P) ).

% case_swap
tff(fact_7506_snd__swap,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),Xb)) = aa(product_prod(A,B),A,product_fst(A,B),Xb) ).

% snd_swap
tff(fact_7507_fst__swap,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(B,A)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),Xb)) = aa(product_prod(B,A),A,product_snd(B,A),Xb) ).

% fst_swap
tff(fact_7508_rotate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,aa(nat,nat,suc,Nb)),Xs) = aa(list(A),list(A),rotate1(A),aa(list(A),list(A),rotate(A,Nb),Xs)) ).

% rotate_Suc
tff(fact_7509_pair__in__swap__image,axiom,
    ! [A: $tType,B: $tType,Yb: A,Xb: B,A5: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Yb),Xb),aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),A5))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Yb),A5) ) ).

% pair_in_swap_image
tff(fact_7510_rotate0,axiom,
    ! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).

% rotate0
tff(fact_7511_surj__swap,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).

% surj_swap
tff(fact_7512_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( aa(list(A),list(A),rotate(A,Nb),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_7513_rotate__id,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),rotate(A,Nb),Xs) = Xs ) ) ).

% rotate_id
tff(fact_7514_rotate__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate(A,Nb),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rotate(B,Nb),Xs)) ).

% rotate_map
tff(fact_7515_rotate__append,axiom,
    ! [A: $tType,L: list(A),Q2: list(A)] : aa(list(A),list(A),rotate(A,aa(list(A),nat,size_size(list(A)),L)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),Q2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Q2),L) ).

% rotate_append
tff(fact_7516_rotate__conv__mod,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Nb),Xs) = aa(list(A),list(A),rotate(A,modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% rotate_conv_mod
tff(fact_7517_inj__swap,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),product_swap(A,B),A5) ).

% inj_swap
tff(fact_7518_rotate1__rotate__swap,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),list(A),rotate(A,Nb),aa(list(A),list(A),rotate1(A),Xs)) ).

% rotate1_rotate_swap
tff(fact_7519_rotate__rotate,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Mb),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),Xs) ).

% rotate_rotate
tff(fact_7520_rotate__def,axiom,
    ! [A: $tType,Nb: nat] : rotate(A,Nb) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(nat,fun(fun(list(A),list(A)),fun(list(A),list(A))),compow(fun(list(A),list(A))),Nb),rotate1(A)) ).

% rotate_def
tff(fact_7521_rotate__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat] : rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),rotate(A,Mb)),rotate(A,Nb)) ).

% rotate_add
tff(fact_7522_product__swap,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B5: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),product_Sigma(B,A,A5,aTP_Lamp_ms(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_ahr(set(B),fun(A,set(B)),A5)) ).

% product_swap
tff(fact_7523_prod_Oswap__def,axiom,
    ! [A: $tType,B: $tType,P: product_prod(B,A)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),P) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(B,A),A,product_snd(B,A),P)),aa(product_prod(B,A),B,product_fst(B,A),P)) ).

% prod.swap_def
tff(fact_7524_rotate__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Nb),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)))),Xs)) ).

% rotate_rev
tff(fact_7525_nth__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,Mb),Xs)),Nb) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_7526_hd__rotate__conv__nth,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% hd_rotate_conv_nth
tff(fact_7527_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,A),G3: fun(C,B),A5: 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_yr(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F3),G3)),A5) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F3),A5),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_ajb(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F3),G3),A5)) ).

% image_split_eq_Sigma
tff(fact_7528_listrel1__subset__listrel,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),R4)
     => ( refl_on(A,top_top(set(A)),R4)
       => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel(A,A,R4)) ) ) ).

% listrel1_subset_listrel
tff(fact_7529_vimage__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : vimage(A,B,F3,bot_bot(set(B))) = bot_bot(set(A)) ).

% vimage_empty
tff(fact_7530_vimage__const,axiom,
    ! [B: $tType,A: $tType,C3: B,A5: set(B)] :
      vimage(A,B,aTP_Lamp_lx(B,fun(A,B),C3),A5) = $ite(member(B,C3,A5),top_top(set(A)),bot_bot(set(A))) ).

% vimage_const
tff(fact_7531_vimage__if,axiom,
    ! [B: $tType,A: $tType,B5: set(A),C3: B,D3: B,A5: set(B)] :
      vimage(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_ajc(set(A),fun(B,fun(B,fun(A,B))),B5),C3),D3),A5) = $ite(
        member(B,C3,A5),
        $ite(member(B,D3,A5),top_top(set(A)),B5),
        $ite(member(B,D3,A5),aa(set(A),set(A),uminus_uminus(set(A)),B5),bot_bot(set(A))) ) ).

% vimage_if
tff(fact_7532_surj__vimage__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => ( ( vimage(B,A,F3,A5) = bot_bot(set(B)) )
      <=> ( A5 = bot_bot(set(A)) ) ) ) ).

% surj_vimage_empty
tff(fact_7533_finite__vimageD,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F4: set(B)] :
      ( aa(set(A),$o,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)) )
       => aa(set(B),$o,finite_finite2(B),F4) ) ) ).

% finite_vimageD
tff(fact_7534_vimage__Suc__insert__0,axiom,
    ! [A5: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),A5)) = vimage(nat,nat,suc,A5) ).

% vimage_Suc_insert_0
tff(fact_7535_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,F3: fun(A,B),B2: B] :
      ( member(A,A2,vimage(A,B,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),B2),bot_bot(set(B)))))
    <=> ( aa(A,B,F3,A2) = B2 ) ) ).

% vimage_singleton_eq
tff(fact_7536_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A2: B,B5: set(B)] : vimage(A,B,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B))))),vimage(A,B,F3,B5)) ).

% vimage_insert
tff(fact_7537_vimage__Suc__insert__Suc,axiom,
    ! [Nb: nat,A5: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Nb)),A5)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Nb),vimage(nat,nat,suc,A5)) ).

% vimage_Suc_insert_Suc
tff(fact_7538_finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( inj_on(B,A,H,A5)
       => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,F4)),A5)) ) ) ).

% finite_vimage_IntI
tff(fact_7539_finite__vimage__Suc__iff,axiom,
    ! [F4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),vimage(nat,nat,suc,F4))
    <=> aa(set(nat),$o,finite_finite2(nat),F4) ) ).

% finite_vimage_Suc_iff
tff(fact_7540_finite__vimageI,axiom,
    ! [B: $tType,A: $tType,F4: set(A),H: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( inj_on(B,A,H,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),vimage(B,A,H,F4)) ) ) ).

% finite_vimageI
tff(fact_7541_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_7542_refl__onD2,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,Yb: A] :
      ( refl_on(A,A5,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
       => member(A,Yb,A5) ) ) ).

% refl_onD2
tff(fact_7543_refl__onD1,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,Yb: A] :
      ( refl_on(A,A5,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
       => member(A,Xb,A5) ) ) ).

% refl_onD1
tff(fact_7544_refl__onD,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),A2: A] :
      ( refl_on(A,A5,R2)
     => ( member(A,A2,A5)
       => 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_7545_vimage__Times,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,product_prod(B,C)),A5: set(B),B5: set(C)] : vimage(A,product_prod(B,C),F3,product_Sigma(B,C,A5,aTP_Lamp_ajd(set(C),fun(B,set(C)),B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),F3),A5)),vimage(A,C,aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F3),B5)) ).

% vimage_Times
tff(fact_7546_vimage__snd,axiom,
    ! [A: $tType,B: $tType,A5: set(B)] : vimage(product_prod(A,B),B,product_snd(A,B),A5) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahr(set(B),fun(A,set(B)),A5)) ).

% vimage_snd
tff(fact_7547_vimage__fst,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : vimage(product_prod(A,B),A,product_fst(A,B),A5) = product_Sigma(A,B,A5,aTP_Lamp_ahq(A,set(B))) ).

% vimage_fst
tff(fact_7548_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,Xb: B,A5: set(B),F3: fun(B,set(A))] :
      vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),product_Sigma(B,A,A5,F3)) = $ite(member(B,Xb,A5),aa(B,set(A),F3,Xb),bot_bot(set(A))) ).

% Pair_vimage_Sigma
tff(fact_7549_finite__vimageD_H,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),vimage(A,B,F3,A5))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),aa(set(A),set(B),image2(A,B,F3),top_top(set(A))))
       => aa(set(B),$o,finite_finite2(B),A5) ) ) ).

% finite_vimageD'
tff(fact_7550_inf__img__fin__dom,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5))
     => ( ~ aa(set(B),$o,finite_finite2(B),A5)
       => ? [X3: A] :
            ( member(A,X3,aa(set(B),set(A),image2(B,A,F3),A5))
            & ~ aa(set(B),$o,finite_finite2(B),vimage(B,A,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_dom
tff(fact_7551_inf__img__fin__domE,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5))
     => ( ~ aa(set(B),$o,finite_finite2(B),A5)
       => ~ ! [Y: A] :
              ( member(A,Y,aa(set(B),set(A),image2(B,A,F3),A5))
             => aa(set(B),$o,finite_finite2(B),vimage(B,A,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_domE
tff(fact_7552_finite__finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ! [Y: A] :
            ( member(A,Y,F4)
           => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))),A5)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,F4)),A5)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_7553_refl__on__def,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A5,R2)
    <=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_aie(set(A),fun(A,set(A)),A5)))
        & ! [X4: A] :
            ( member(A,X4,A5)
           => 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_def
tff(fact_7554_refl__onI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_aie(set(A),fun(A,set(A)),A5)))
     => ( ! [X3: A] :
            ( member(A,X3,A5)
           => 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(A,A5,R2) ) ) ).

% refl_onI
tff(fact_7555_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),B5: set(B)] : vimage(A,B,F3,B5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aje(fun(A,B),fun(B,set(A)),F3)),B5)) ).

% vimage_eq_UN
tff(fact_7556_refl__on__def_H,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A5,R2)
    <=> ( ! [X4: product_prod(A,A)] :
            ( member(product_prod(A,A),X4,R2)
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_ajf(set(A),fun(A,fun(A,$o)),A5)),X4) )
        & ! [X4: A] :
            ( member(A,X4,A5)
           => 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_def'
tff(fact_7557_inf__img__fin__dom_H,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5))
     => ( ~ aa(set(B),$o,finite_finite2(B),A5)
       => ? [X3: A] :
            ( member(A,X3,aa(set(B),set(A),image2(B,A,F3),A5))
            & ~ aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))),A5)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_7558_inf__img__fin__domE_H,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F3),A5))
     => ( ~ aa(set(B),$o,finite_finite2(B),A5)
       => ~ ! [Y: A] :
              ( member(A,Y,aa(set(B),set(A),image2(B,A,F3),A5))
             => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))),A5)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_7559_card__vimage__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),aa(set(A),set(B),image2(A,B,F3),top_top(set(A))))
       => ( aa(set(A),nat,finite_card(A),vimage(A,B,F3,A5)) = aa(set(B),nat,finite_card(B),A5) ) ) ) ).

% card_vimage_inj
tff(fact_7560_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),D5: set(A),A5: set(B)] :
      ( inj_on(A,B,F3,D5)
     => ( aa(set(B),$o,finite_finite2(B),A5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F3,A5)),D5))),aa(set(B),nat,finite_card(B),A5)) ) ) ).

% card_vimage_inj_on_le
tff(fact_7561_set__decode__div__2,axiom,
    ! [Xb: nat] : nat_set_decode(divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),bit0(one2)))) = vimage(nat,nat,suc,nat_set_decode(Xb)) ).

% set_decode_div_2
tff(fact_7562_set__encode__vimage__Suc,axiom,
    ! [A5: set(nat)] : aa(set(nat),nat,nat_set_encode,vimage(nat,nat,suc,A5)) = divide_divide(nat,aa(set(nat),nat,nat_set_encode,A5),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% set_encode_vimage_Suc
tff(fact_7563_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A2: B] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the(A,aa(B,fun(A,$o),aTP_Lamp_ajg(fun(A,B),fun(B,fun(A,$o)),F3),A2))),bot_bot(set(A)))) ) ).

% inj_vimage_singleton
tff(fact_7564_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),A2: B] :
      ( inj_on(A,B,F3,A5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B))))),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_ajh(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F3),A5),A2))),bot_bot(set(A)))) ) ).

% inj_on_vimage_singleton
tff(fact_7565_refl__on__singleton,axiom,
    ! [A: $tType,Xb: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_7566_inv__image__partition,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o),Ys: list(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X3) )
     => ( ! [Y: A] :
            ( member(A,Y,aa(list(A),set(A),set2(A),Ys))
           => ~ aa(A,$o,P2,Y) )
       => ( vimage(list(A),product_prod(list(A),list(A)),partition(A,P2),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)))),insert(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_7567_refl__on__domain,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( refl_on(A,A5,R2)
     => ( 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)
       => ( member(A,A2,A5)
          & member(A,B2,A5) ) ) ) ).

% refl_on_domain
tff(fact_7568_Restr__natLeq,axiom,
    ! [Nb: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb)),aTP_Lamp_aji(nat,fun(nat,set(nat)),Nb))) = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ajj(nat,fun(nat,fun(nat,$o)),Nb))) ).

% Restr_natLeq
tff(fact_7569_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less_eq(nat))) ).

% natLeq_def
tff(fact_7570_Restr__natLeq2,axiom,
    ! [Nb: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,Nb),aTP_Lamp_ajk(nat,fun(nat,set(nat)),Nb))) = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ajj(nat,fun(nat,fun(nat,$o)),Nb))) ).

% Restr_natLeq2
tff(fact_7571_linear__order__on__singleton,axiom,
    ! [A: $tType,Xb: A] : order_679001287576687338der_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_7572_underS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : order_underS(A,R2,A2) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ajl(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A2)) ).

% underS_def
tff(fact_7573_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_7574_natLeq__underS__less,axiom,
    ! [Nb: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,Nb) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb)) ).

% natLeq_underS_less
tff(fact_7575_underS__I,axiom,
    ! [A: $tType,I2: A,J: A,R: set(product_prod(A,A))] :
      ( ( I2 != J )
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J),R)
       => member(A,I2,order_underS(A,R,J)) ) ) ).

% underS_I
tff(fact_7576_underS__E,axiom,
    ! [A: $tType,I2: A,R: set(product_prod(A,A)),J: A] :
      ( member(A,I2,order_underS(A,R,J))
     => ( ( I2 != J )
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J),R) ) ) ).

% underS_E
tff(fact_7577_total__on__singleton,axiom,
    ! [A: $tType,Xb: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_7578_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_7579_total__pair__less,axiom,
    ! [A5: set(product_prod(nat,nat))] : total_on(product_prod(nat,nat),A5,fun_pair_less) ).

% total_pair_less
tff(fact_7580_total__on__def,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A5,R2)
    <=> ! [X4: A] :
          ( member(A,X4,A5)
         => ! [Xa2: A] :
              ( member(A,Xa2,A5)
             => ( ( X4 != Xa2 )
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R2)
                  | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_7581_total__onI,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,A5)
         => ( member(A,Y,A5)
           => ( ( X3 != Y )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),R2)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X3),R2) ) ) ) )
     => total_on(A,A5,R2) ) ).

% total_onI
tff(fact_7582_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_7583_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_7584_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_7585_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( sgn_sgn(complex,Z) = cis(arg(Z)) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_7586_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)))) ) ).

% slice_eq_mask
tff(fact_7587_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) ) ).

% not_negative_int_iff
tff(fact_7588_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_7589_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_7590_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_7591_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_7592_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_7593_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_diff_distrib
tff(fact_7594_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_add_distrib
tff(fact_7595_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_7596_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_7597_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),bit_se2239418461657761734s_mask(A,Nb)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Mb))) ) ).

% push_bit_mask_eq
tff(fact_7598_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Yb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Yb) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Yb) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) = Yb ) ) ) ) ).

% bit.compl_unique
tff(fact_7599_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Nb)
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) )
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_not_iff_eq
tff(fact_7600_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( sgn_sgn(complex,Z) = cis(Xb) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => ( arg(Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_7601_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => bij_betw(nat,complex,aTP_Lamp_ajm(nat,fun(nat,complex),Nb),aa(nat,set(nat),set_ord_lessThan(nat),Nb),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_cn(nat,fun(complex,$o),Nb))) ) ).

% bij_betw_roots_unity
tff(fact_7602_bij__betw__nth__root__unity,axiom,
    ! [C3: complex,Nb: nat] :
      ( ( C3 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,root(Nb),real_V7770717601297561774m_norm(complex,C3)))),cis(divide_divide(real,arg(C3),aa(nat,real,semiring_1_of_nat(real),Nb))))),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_cn(nat,fun(complex,$o),Nb)),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_jt(complex,fun(nat,fun(complex,$o)),C3),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_7603_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A5: set(A),B5: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A2),A5,B5)
        <=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),A5) = B5 ) ) ) ).

% bij_betw_add
tff(fact_7604_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N3: set(nat),A5: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N3,A5)
        <=> ( aa(set(nat),set(A),image2(nat,A,semiring_1_of_nat(A)),N3) = A5 ) ) ) ).

% bij_betw_of_nat
tff(fact_7605_bij__betw__funpow,axiom,
    ! [A: $tType,F3: fun(A,A),S2: set(A),Nb: nat] :
      ( bij_betw(A,A,F3,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)),Nb),F3),S2,S2) ) ).

% bij_betw_funpow
tff(fact_7606_bij__fn,axiom,
    ! [A: $tType,F3: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F3,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_7607_bij__betw__same__card,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(B)] :
      ( bij_betw(A,B,F3,A5,B5)
     => ( aa(set(A),nat,finite_card(A),A5) = aa(set(B),nat,finite_card(B),B5) ) ) ).

% bij_betw_same_card
tff(fact_7608_bij__betw__iff__card,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( ? [F7: fun(A,B)] : bij_betw(A,B,F7,A5,B5)
        <=> ( aa(set(A),nat,finite_card(A),A5) = aa(set(B),nat,finite_card(B),B5) ) ) ) ) ).

% bij_betw_iff_card
tff(fact_7609_finite__same__card__bij,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( ( aa(set(A),nat,finite_card(A),A5) = aa(set(B),nat,finite_card(B),B5) )
         => ? [H4: fun(A,B)] : bij_betw(A,B,H4,A5,B5) ) ) ) ).

% finite_same_card_bij
tff(fact_7610_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B)] :
      ( bij_betw(A,B,F3,bot_bot(set(A)),A5)
     => ( A5 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_7611_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( bij_betw(A,B,F3,A5,bot_bot(set(B)))
     => ( A5 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_7612_bij__betw__finite,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(B)] :
      ( bij_betw(A,B,F3,A5,B5)
     => ( aa(set(A),$o,finite_finite2(A),A5)
      <=> aa(set(B),$o,finite_finite2(B),B5) ) ) ).

% bij_betw_finite
tff(fact_7613_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),C5: set(B),G3: fun(A,B),B5: set(A),D5: set(B)] :
      ( bij_betw(A,B,F3,A5,C5)
     => ( bij_betw(A,B,G3,B5,D5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C5),D5) = 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_abc(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F3),A5),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C5),D5)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_7614_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A5: set(A),F3: fun(A,B),A10: set(B)] :
      ( ~ member(A,B2,A5)
     => ( ~ member(B,aa(A,B,F3,B2),A10)
       => ( bij_betw(A,B,F3,A5,A10)
         => bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A10),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_7615_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A5: set(A),F3: fun(A,B),A10: set(B)] :
      ( ~ member(A,B2,A5)
     => ( ~ member(B,aa(A,B,F3,B2),A10)
       => ( bij_betw(A,B,F3,A5,A10)
        <=> bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A10),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_7616_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),C5: set(A),B5: set(B),D5: set(B)] :
      ( bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B5),D5))
     => ( bij_betw(A,B,F3,C5,D5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C5) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B5),D5) = bot_bot(set(B)) )
           => bij_betw(A,B,F3,A5,B5) ) ) ) ) ).

% bij_betw_partition
tff(fact_7617_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(B),C5: set(A),D5: set(B)] :
      ( bij_betw(A,B,F3,A5,B5)
     => ( bij_betw(A,B,F3,C5,D5)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B5),D5) = bot_bot(set(B)) )
         => bij_betw(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B5),D5)) ) ) ) ).

% bij_betw_combine
tff(fact_7618_finite__vimage__iff,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F4: set(B)] :
      ( bij_betw(A,B,H,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),$o,finite_finite2(A),vimage(A,B,H,F4))
      <=> aa(set(B),$o,finite_finite2(B),F4) ) ) ).

% finite_vimage_iff
tff(fact_7619_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A5: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ? [H4: fun(A,A)] : bij_betw(A,A,H4,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_7620_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A5: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ? [H4: fun(A,A)] : bij_betw(A,A,H4,A5,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)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_7621_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M4)
     => ? [H4: fun(nat,A)] : bij_betw(nat,A,H4,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M4)),M4) ) ).

% ex_bij_betw_nat_finite
tff(fact_7622_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M4)
     => ? [H4: fun(nat,A)] : bij_betw(nat,A,H4,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M4)),M4) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_7623_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T3: set(B),H: fun(A,B),S2: set(A),T2: set(B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
             => ( ! [A4: A] :
                    ( member(A,A4,S5)
                   => ( aa(B,C,G3,aa(A,B,H,A4)) = zero_zero(C) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T3)
                     => ( aa(B,C,G3,B4) = zero_zero(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ajn(fun(A,B),fun(fun(B,C),fun(A,C)),H),G3)),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G3),T2) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_7624_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T3: set(B),H: fun(A,B),S2: set(A),T2: set(B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
             => ( ! [A4: A] :
                    ( member(A,A4,S5)
                   => ( aa(B,C,G3,aa(A,B,H,A4)) = one_one(C) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T3)
                     => ( aa(B,C,G3,B4) = one_one(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ajo(fun(A,B),fun(fun(B,C),fun(A,C)),H),G3)),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G3),T2) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_7625_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat),B5: set(A)] :
      ( distinct(A,Xs)
     => ( ( A5 = aa(nat,set(nat),set_ord_lessThan(nat),aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B5 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A5,B5) ) ) ) ).

% bij_betw_nth
tff(fact_7626_ex__bij__betw__strict__mono__card,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),M4)
         => ~ ! [H4: fun(nat,A)] :
                ( bij_betw(nat,A,H4,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),M4)),M4)
               => ~ strict_mono_on(nat,A,H4,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),M4))) ) ) ) ).

% ex_bij_betw_strict_mono_card
tff(fact_7627_sum_OatLeastAtMost__reindex,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add(B)
        & ord(A) )
     => ! [H: fun(nat,A),Mb: nat,Nb: nat,G3: fun(A,B)] :
          ( bij_betw(nat,A,H,set_or1337092689740270186AtMost(nat,Mb,Nb),set_or1337092689740270186AtMost(A,aa(nat,A,H,Mb),aa(nat,A,H,Nb)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),set_or1337092689740270186AtMost(A,aa(nat,A,H,Mb),aa(nat,A,H,Nb))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,A),fun(nat,B),comp(A,B,nat,G3),H)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ) ) ).

% sum.atLeastAtMost_reindex
tff(fact_7628_sum_OatLeastLessThan__reindex,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add(B)
        & ord(A) )
     => ! [H: fun(nat,A),Mb: nat,Nb: nat,G3: fun(A,B)] :
          ( bij_betw(nat,A,H,set_or7035219750837199246ssThan(nat,Mb,Nb),set_or7035219750837199246ssThan(A,aa(nat,A,H,Mb),aa(nat,A,H,Nb)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),set_or7035219750837199246ssThan(A,aa(nat,A,H,Mb),aa(nat,A,H,Nb))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),aa(fun(nat,A),fun(nat,B),comp(A,B,nat,G3),H)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ) ) ).

% sum.atLeastLessThan_reindex
tff(fact_7629_prod_OatLeastAtMost__reindex,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ord(A) )
     => ! [H: fun(nat,A),Mb: nat,Nb: nat,G3: fun(A,B)] :
          ( bij_betw(nat,A,H,set_or1337092689740270186AtMost(nat,Mb,Nb),set_or1337092689740270186AtMost(A,aa(nat,A,H,Mb),aa(nat,A,H,Nb)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),set_or1337092689740270186AtMost(A,aa(nat,A,H,Mb),aa(nat,A,H,Nb))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,A),fun(nat,B),comp(A,B,nat,G3),H)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ) ) ).

% prod.atLeastAtMost_reindex
tff(fact_7630_prod_OatLeastLessThan__reindex,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ord(A) )
     => ! [H: fun(nat,A),Mb: nat,Nb: nat,G3: fun(A,B)] :
          ( bij_betw(nat,A,H,set_or7035219750837199246ssThan(nat,Mb,Nb),set_or7035219750837199246ssThan(A,aa(nat,A,H,Mb),aa(nat,A,H,Nb)))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),set_or7035219750837199246ssThan(A,aa(nat,A,H,Mb),aa(nat,A,H,Nb))) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7121269368397514597t_prod(nat,B),aa(fun(nat,A),fun(nat,B),comp(A,B,nat,G3),H)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ) ) ).

% prod.atLeastLessThan_reindex
tff(fact_7631_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_ajp(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_7632_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_7633_bij__betw__Suc,axiom,
    ! [M4: set(nat),N3: set(nat)] :
      ( bij_betw(nat,nat,suc,M4,N3)
    <=> ( aa(set(nat),set(nat),image2(nat,nat,suc),M4) = N3 ) ) ).

% bij_betw_Suc
tff(fact_7634_bij__swap,axiom,
    ! [A: $tType,B: $tType] : bij_betw(product_prod(A,B),product_prod(B,A),product_swap(A,B),top_top(set(product_prod(A,B))),top_top(set(product_prod(B,A)))) ).

% bij_swap
tff(fact_7635_some__in__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( member(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,$o),A5)),A5)
    <=> ( A5 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_7636_bij__int__decode,axiom,
    bij_betw(nat,int,nat_int_decode,top_top(set(nat)),top_top(set(int))) ).

% bij_int_decode
tff(fact_7637_bij__int__encode,axiom,
    bij_betw(int,nat,nat_int_encode,top_top(set(int)),top_top(set(nat))) ).

% bij_int_encode
tff(fact_7638_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M4)
     => ? [H4: fun(A,nat)] : bij_betw(A,nat,H4,M4,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M4))) ) ).

% ex_bij_betw_finite_nat
tff(fact_7639_bij__list__encode,axiom,
    bij_betw(list(nat),nat,nat_list_encode,top_top(set(list(nat))),top_top(set(nat))) ).

% bij_list_encode
tff(fact_7640_bij__prod__encode,axiom,
    bij_betw(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat))),top_top(set(nat))) ).

% bij_prod_encode
tff(fact_7641_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( lattic7623131987881927897min_on(A,B,F3,S2) = fChoice(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajq(set(A),fun(fun(A,B),fun(A,$o)),S2),F3)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_7642_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_7643_max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : max_ext(A,R) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_ajr(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R))) ).

% max_ext_eq
tff(fact_7644_bex__empty,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ~ ? [X: A] :
          ( member(A,X,bot_bot(set(A)))
          & aa(A,$o,P2,X) ) ).

% bex_empty
tff(fact_7645_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,Xb: A,Yb: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_it(A,fun(B,fun(A,fun(B,$o))),Xb),Yb))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Yb) ).

% Eps_case_prod_eq
tff(fact_7646_finite__Collect__bex,axiom,
    ! [B: $tType,A: $tType,A5: set(A),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_ajs(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A5),Q)))
      <=> ! [X4: A] :
            ( member(A,X4,A5)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_zx(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),X4))) ) ) ) ).

% finite_Collect_bex
tff(fact_7647_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),$o)] : fChoice(product_prod(A,B),P2) = fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_ajt(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P2))) ).

% split_paired_Eps
tff(fact_7648_Bex__fold,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ? [X4: A] :
            ( member(A,X4,A5)
            & aa(A,$o,P2,X4) )
      <=> finite_fold(A,$o,aTP_Lamp_aju(fun(A,$o),fun(A,fun($o,$o)),P2),$false,A5) ) ) ).

% Bex_fold
tff(fact_7649_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat),B5: set(nat)] : nths(A,nths(A,Xs,A5),B5) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_ajw(set(nat),fun(set(nat),fun(nat,$o)),A5),B5))) ).

% nths_nths
tff(fact_7650_max__extp_Omax__extI,axiom,
    ! [A: $tType,X5: set(A),Y5: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),X5)
     => ( aa(set(A),$o,finite_finite2(A),Y5)
       => ( ( Y5 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => ? [Xa: A] :
                    ( member(A,Xa,Y5)
                    & aa(A,$o,aa(A,fun(A,$o),R,X3),Xa) ) )
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),X5),Y5) ) ) ) ) ).

% max_extp.max_extI
tff(fact_7651_max__extp_Osimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),A1),A22)
    <=> ( aa(set(A),$o,finite_finite2(A),A1)
        & aa(set(A),$o,finite_finite2(A),A22)
        & ( A22 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
        & ! [X4: A] :
            ( member(A,X4,A1)
           => ? [Xa2: A] :
                ( member(A,Xa2,A22)
                & aa(A,$o,aa(A,fun(A,$o),R,X4),Xa2) ) ) ) ) ).

% max_extp.simps
tff(fact_7652_max__extp_Ocases,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R),A1),A22)
     => ~ ( aa(set(A),$o,finite_finite2(A),A1)
         => ( aa(set(A),$o,finite_finite2(A),A22)
           => ( ( A22 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
             => ~ ! [X: A] :
                    ( member(A,X,A1)
                   => ? [Xa3: A] :
                        ( member(A,Xa3,A22)
                        & aa(A,$o,aa(A,fun(A,$o),R,X),Xa3) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_7653_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_7654_min__ext__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : min_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_ajx(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R2)) ).

% min_ext_def
tff(fact_7655_map__project__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),A5: set(B)] : map_project(B,A,F3,A5) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ajy(fun(B,option(A)),fun(set(B),fun(A,$o)),F3),A5)) ).

% map_project_def
tff(fact_7656_prod_Oset__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(list(B),set(B),set2(B),Xs)) = groups5270119922927024881d_list(A,aa(list(B),list(A),map(B,A,G3),remdups(B,Xs))) ) ).

% prod.set_conv_list
tff(fact_7657_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xs: list(A)] :
          ( ( groups5270119922927024881d_list(A,Xs) = zero_zero(A) )
        <=> member(A,zero_zero(A),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% prod_list_zero_iff
tff(fact_7658_prod_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Xs: list(A),G3: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(list(A),set(A),set2(A),Xs)) = groups5270119922927024881d_list(B,aa(list(A),list(B),map(A,B,G3),Xs)) ) ) ) ).

% prod.distinct_set_conv_list
tff(fact_7659_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))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% lists_empty
tff(fact_7660_card__quotient__disjoint,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( inj_on(A,set(set(A)),aTP_Lamp_ajz(set(product_prod(A,A)),fun(A,set(set(A))),R2),A5)
       => ( aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A5,R2)) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_quotient_disjoint
tff(fact_7661_Cons__in__lists__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),A5: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),lists(A,A5))
    <=> ( member(A,Xb,A5)
        & member(list(A),Xs,lists(A,A5)) ) ) ).

% Cons_in_lists_iff
tff(fact_7662_in__listsI,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => member(A,X3,A5) )
     => member(list(A),Xs,lists(A,A5)) ) ).

% in_listsI
tff(fact_7663_lists__Int__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = 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,A5)),lists(A,B5)) ).

% lists_Int_eq
tff(fact_7664_append__in__lists__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),A5: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),lists(A,A5))
    <=> ( member(list(A),Xs,lists(A,A5))
        & member(list(A),Ys,lists(A,A5)) ) ) ).

% append_in_lists_conv
tff(fact_7665_lists__UNIV,axiom,
    ! [A: $tType] : lists(A,top_top(set(A))) = top_top(set(list(A))) ).

% lists_UNIV
tff(fact_7666_listrel__refl__on,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A5,R2)
     => refl_on(list(A),lists(A,A5),listrel(A,A,R2)) ) ).

% listrel_refl_on
tff(fact_7667_lists__eq__set,axiom,
    ! [A: $tType,A5: set(A)] : lists(A,A5) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_aka(set(A),fun(list(A),$o),A5)) ).

% lists_eq_set
tff(fact_7668_in__lists__conv__set,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] :
      ( member(list(A),Xs,lists(A,A5))
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => member(A,X4,A5) ) ) ).

% in_lists_conv_set
tff(fact_7669_in__listsD,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] :
      ( member(list(A),Xs,lists(A,A5))
     => ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => member(A,X,A5) ) ) ).

% in_listsD
tff(fact_7670_lists_ONil,axiom,
    ! [A: $tType,A5: set(A)] : member(list(A),nil(A),lists(A,A5)) ).

% lists.Nil
tff(fact_7671_lists_Ocases,axiom,
    ! [A: $tType,A2: list(A),A5: set(A)] :
      ( member(list(A),A2,lists(A,A5))
     => ( ( 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) )
             => ( member(A,A4,A5)
               => ~ member(list(A),L3,lists(A,A5)) ) ) ) ) ).

% lists.cases
tff(fact_7672_lists_Osimps,axiom,
    ! [A: $tType,A2: list(A),A5: set(A)] :
      ( member(list(A),A2,lists(A,A5))
    <=> ( ( 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) )
            & member(A,A7,A5)
            & member(list(A),L4,lists(A,A5)) ) ) ) ).

% lists.simps
tff(fact_7673_listsE,axiom,
    ! [A: $tType,Xb: A,L: list(A),A5: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),L),lists(A,A5))
     => ~ ( member(A,Xb,A5)
         => ~ member(list(A),L,lists(A,A5)) ) ) ).

% listsE
tff(fact_7674_lists_OCons,axiom,
    ! [A: $tType,A2: A,A5: set(A),L: list(A)] :
      ( member(A,A2,A5)
     => ( member(list(A),L,lists(A,A5))
       => member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),L),lists(A,A5)) ) ) ).

% lists.Cons
tff(fact_7675_lists__IntI,axiom,
    ! [A: $tType,L: list(A),A5: set(A),B5: set(A)] :
      ( member(list(A),L,lists(A,A5))
     => ( member(list(A),L,lists(A,B5))
       => member(list(A),L,lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ).

% lists_IntI
tff(fact_7676_lists__mono,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),lists(A,A5)),lists(A,B5)) ) ).

% lists_mono
tff(fact_7677_lists__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] : lists(A,aa(set(B),set(A),image2(B,A,F3),A5)) = aa(set(list(B)),set(list(A)),image2(list(B),list(A),map(B,A,F3)),lists(B,A5)) ).

% lists_image
tff(fact_7678_listrel__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_aie(set(A),fun(A,set(A)),A5)))
     => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R2)),product_Sigma(list(A),list(A),lists(A,A5),aTP_Lamp_akb(set(A),fun(list(A),set(list(A))),A5))) ) ).

% listrel_subset
tff(fact_7679_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_7680_quotient__is__empty,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( ( equiv_quotient(A,A5,R2) = bot_bot(set(set(A))) )
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% quotient_is_empty
tff(fact_7681_quotient__is__empty2,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( ( bot_bot(set(set(A))) = equiv_quotient(A,A5,R2) )
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% quotient_is_empty2
tff(fact_7682_finite__equiv__class,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_aie(set(A),fun(A,set(A)),A5)))
       => ( member(set(A),X5,equiv_quotient(A,A5,R2))
         => aa(set(A),$o,finite_finite2(A),X5) ) ) ) ).

% finite_equiv_class
tff(fact_7683_finite__quotient,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_aie(set(A),fun(A,set(A)),A5)))
       => aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A5,R2)) ) ) ).

% finite_quotient
tff(fact_7684_quotient__diff1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A),A2: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_ajz(set(product_prod(A,A)),fun(A,set(set(A))),R2),A5)
     => ( member(A,A2,A5)
       => ( equiv_quotient(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)),insert(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,A5,R2)),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),R2)) ) ) ) ).

% quotient_diff1
tff(fact_7685_Collect__finite__subset__eq__lists,axiom,
    ! [A: $tType,T2: set(A)] : aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aco(set(A),fun(set(A),$o),T2)) = aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),lists(A,T2)) ).

% Collect_finite_subset_eq_lists
tff(fact_7686_Collect__finite__eq__lists,axiom,
    ! [A: $tType] : aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A)) = aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),lists(A,top_top(set(A)))) ).

% Collect_finite_eq_lists
tff(fact_7687_to__nat__on__def,axiom,
    ! [A: $tType,S2: set(A)] : countable_to_nat_on(A,S2) = fChoice(fun(A,nat),aTP_Lamp_akc(set(A),fun(fun(A,nat),$o),S2)) ).

% to_nat_on_def
tff(fact_7688_to__nat__on__finite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => bij_betw(A,nat,countable_to_nat_on(A,S2),S2,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),S2))) ) ).

% to_nat_on_finite
tff(fact_7689_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F3),Xs),top_top(A)) ) ).

% INF_set_fold
tff(fact_7690_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F3),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F3),Xs),bot_bot(A)) ) ).

% SUP_set_fold
tff(fact_7691_fold__append,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xs: list(B),Ys: list(B)] : fold(B,A,F3,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,fold(B,A,F3,Ys)),fold(B,A,F3,Xs)) ).

% fold_append
tff(fact_7692_fold__replicate,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Nb: nat,Xb: B] : fold(B,A,F3,replicate(B,Nb,Xb)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(B,fun(A,A),F3,Xb)) ).

% fold_replicate
tff(fact_7693_fold__Cons__rev,axiom,
    ! [A: $tType,Xs: 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_7694_rev__conv__fold,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),fold(A,list(A),cons(A),Xs),nil(A)) ).

% rev_conv_fold
tff(fact_7695_fold__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: fun(B,fun(A,A)),F3: fun(C,B),Xs: list(C)] : fold(B,A,G3,aa(list(C),list(B),map(C,B,F3),Xs)) = fold(C,A,aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G3),F3),Xs) ).

% fold_map
tff(fact_7696_fold__commute__apply,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G3: fun(A,fun(B,B)),F3: fun(A,fun(C,C)),S: B] :
      ( ! [X3: A] :
          ( member(A,X3,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),G3,X3)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F3,X3)),H) ) )
     => ( aa(B,C,H,aa(B,B,fold(A,B,G3,Xs),S)) = aa(C,C,fold(A,C,F3,Xs),aa(B,C,H,S)) ) ) ).

% fold_commute_apply
tff(fact_7697_fold__commute,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G3: fun(A,fun(B,B)),F3: fun(A,fun(C,C))] :
      ( ! [X3: A] :
          ( member(A,X3,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),G3,X3)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F3,X3)),H) ) )
     => ( aa(fun(B,B),fun(B,C),comp(B,C,B,H),fold(A,B,G3,Xs)) = aa(fun(B,C),fun(B,C),comp(C,C,B,fold(A,C,F3,Xs)),H) ) ) ).

% fold_commute
tff(fact_7698_fold__Cons,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xb: B,Xs: list(B)] : fold(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)) = aa(fun(A,A),fun(A,A),comp(A,A,A,fold(B,A,F3,Xs)),aa(B,fun(A,A),F3,Xb)) ).

% fold_Cons
tff(fact_7699_foldr__conv__fold,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xs: list(B)] : foldr(B,A,F3,Xs) = fold(B,A,F3,aa(list(B),list(B),rev(B),Xs)) ).

% foldr_conv_fold
tff(fact_7700_fold__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P2: fun(B,$o),Xs: list(B)] : fold(B,A,F3,aa(list(B),list(B),filter2(B,P2),Xs)) = fold(B,A,aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_agz(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F3),P2),Xs) ).

% fold_filter
tff(fact_7701_fold__invariant,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Q: fun(A,$o),P2: fun(B,$o),S: B,F3: fun(A,fun(B,B))] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,Q,X3) )
     => ( aa(B,$o,P2,S)
       => ( ! [X3: A,S3: B] :
              ( aa(A,$o,Q,X3)
             => ( aa(B,$o,P2,S3)
               => aa(B,$o,P2,aa(B,B,aa(A,fun(B,B),F3,X3),S3)) ) )
         => aa(B,$o,P2,aa(B,B,fold(A,B,F3,Xs),S)) ) ) ) ).

% fold_invariant
tff(fact_7702_List_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,Xs: list(B),Ys: list(B),F3: fun(B,fun(A,A)),G3: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X3: B] :
              ( member(B,X3,aa(list(B),set(B),set2(B),Xs))
             => ( aa(B,fun(A,A),F3,X3) = aa(B,fun(A,A),G3,X3) ) )
         => ( aa(A,A,fold(B,A,F3,Xs),A2) = aa(A,A,fold(B,A,G3,Ys),B2) ) ) ) ) ).

% List.fold_cong
tff(fact_7703_fold__id,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,fun(B,B),F3,X3) = id(B) ) )
     => ( fold(A,B,F3,Xs) = id(B) ) ) ).

% fold_id
tff(fact_7704_fold__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),S: A] : aa(A,A,fold(B,A,F3,nil(B)),S) = S ).

% fold_simps(1)
tff(fact_7705_fold__Nil,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A))] : fold(B,A,F3,nil(B)) = id(A) ).

% fold_Nil
tff(fact_7706_fold__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),Xb: B,Xs: list(B),S: A] : aa(A,A,fold(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)),S) = aa(A,A,fold(B,A,F3,Xs),aa(A,A,aa(B,fun(A,A),F3,Xb),S)) ).

% fold_simps(2)
tff(fact_7707_foldl__conv__fold,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),S: A,Xs: list(B)] : foldl(A,B,F3,S,Xs) = aa(A,A,fold(B,A,aTP_Lamp_aha(fun(A,fun(B,A)),fun(B,fun(A,A)),F3),Xs),S) ).

% foldl_conv_fold
tff(fact_7708_union__set__fold,axiom,
    ! [A: $tType,Xs: list(A),A5: 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)),A5) = aa(set(A),set(A),fold(A,set(A),insert(A),Xs),A5) ).

% union_set_fold
tff(fact_7709_fold__rev,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y)),aa(A,fun(B,B),F3,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y)) ) ) )
     => ( fold(A,B,F3,aa(list(A),list(A),rev(A),Xs)) = fold(A,B,F3,Xs) ) ) ).

% fold_rev
tff(fact_7710_fold__plus__sum__list__rev,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : fold(A,A,plus_plus(A),Xs) = aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(A),list(A),rev(A),Xs))) ) ).

% fold_plus_sum_list_rev
tff(fact_7711_fold__remove1__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,fun(B,B)),Xb: A] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y)),aa(A,fun(B,B),F3,X3)) ) ) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( fold(A,B,F3,Xs) = aa(fun(B,B),fun(B,B),comp(B,B,B,fold(A,B,F3,remove1(A,Xb,Xs))),aa(A,fun(B,B),F3,Xb)) ) ) ) ).

% fold_remove1_split
tff(fact_7712_foldr__fold,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y)),aa(A,fun(B,B),F3,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y)) ) ) )
     => ( foldr(A,B,F3,Xs) = fold(A,B,F3,Xs) ) ) ).

% foldr_fold
tff(fact_7713_minus__set__fold,axiom,
    ! [A: $tType,A5: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(list(A),set(A),set2(A),Xs)) = aa(set(A),set(A),fold(A,set(A),remove(A),Xs),A5) ).

% minus_set_fold
tff(fact_7714_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : gcd_Gcd(int,aa(list(int),set(int),set2(int),Xs)) = aa(int,int,fold(int,int,gcd_gcd(int),Xs),zero_zero(int)) ).

% Gcd_int_set_eq_fold
tff(fact_7715_fold__append__concat__rev,axiom,
    ! [A: $tType,Xss: list(list(A))] : fold(list(A),list(A),append(A),Xss) = aa(list(A),fun(list(A),list(A)),append(A),concat(A,aa(list(list(A)),list(list(A)),rev(list(A)),Xss))) ).

% fold_append_concat_rev
tff(fact_7716_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,fold(A,A,sup_sup(A),Xs),bot_bot(A)) ) ).

% Sup_set_fold
tff(fact_7717_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,fold(A,A,inf_inf(A),Xs),top_top(A)) ) ).

% Inf_set_fold
tff(fact_7718_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,fold(A,A,gcd_gcd(A),Xs),zero_zero(A)) ) ).

% Gcd_set_eq_fold
tff(fact_7719_Inf__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: 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),Xb),Xs))) = aa(A,A,fold(A,A,inf_inf(A),Xs),Xb) ) ).

% Inf_fin.set_eq_fold
tff(fact_7720_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: 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),Xb),Xs))) = aa(A,A,fold(A,A,sup_sup(A),Xs),Xb) ) ).

% Sup_fin.set_eq_fold
tff(fact_7721_Max_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: 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),Xb),Xs))) = aa(A,A,fold(A,A,ord_max(A),Xs),Xb) ) ).

% Max.set_eq_fold
tff(fact_7722_Min_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: 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),Xb),Xs))) = aa(A,A,fold(A,A,ord_min(A),Xs),Xb) ) ).

% Min.set_eq_fold
tff(fact_7723_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xs: list(A),Yb: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S2)
       => ( finite_fold(A,B,F3,Yb,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F3,Xs),Yb) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_7724_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,fold(nat,nat,gcd_gcd(nat),Xs),zero_zero(nat)) ).

% Gcd_nat_set_eq_fold
tff(fact_7725_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xs: list(A),Yb: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S2)
       => ( finite_fold(A,B,F3,Yb,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F3,remdups(A,Xs)),Yb) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_7726_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X5: set(A)] :
      ( countable_countable(A,X5)
     => ~ ! [F5: fun(nat,set(A))] :
            ( ! [I: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F5,I)),X5)
           => ( ! [I: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F5,I)),aa(nat,set(A),F5,aa(nat,nat,suc,I)))
             => ( ! [I: nat] : aa(set(A),$o,finite_finite2(A),aa(nat,set(A),F5,I))
               => ( 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)))) != X5 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_7727_butlast__power,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(nat,fun(fun(list(A),list(A)),fun(list(A),list(A))),compow(fun(list(A),list(A))),Nb),butlast(A)),Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs) ).

% butlast_power
tff(fact_7728_countable__empty,axiom,
    ! [A: $tType] : countable_countable(A,bot_bot(set(A))) ).

% countable_empty
tff(fact_7729_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_7730_countable__Diff__eq,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( countable_countable(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)),insert(A),Xb),bot_bot(set(A)))))
    <=> countable_countable(A,A5) ) ).

% countable_Diff_eq
tff(fact_7731_butlast__snoc,axiom,
    ! [A: $tType,Xs: list(A),Xb: 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),Xb),nil(A)))) = Xs ).

% butlast_snoc
tff(fact_7732_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_7733_countable__Collect__finite__subset,axiom,
    ! [A: $tType,T2: set(A)] :
      ( countable_countable(A,T2)
     => countable_countable(set(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aco(set(A),fun(set(A),$o),T2))) ) ).

% countable_Collect_finite_subset
tff(fact_7734_infinite__countable__subset_H,axiom,
    ! [A: $tType,X5: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),X5)
     => ? [C6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C6),X5)
          & countable_countable(A,C6)
          & ~ aa(set(A),$o,finite_finite2(A),C6) ) ) ).

% infinite_countable_subset'
tff(fact_7735_countable__Collect__finite,axiom,
    ! [A: $tType] :
      ( countable(A)
     => countable_countable(set(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A))) ) ).

% countable_Collect_finite
tff(fact_7736_uncountable__infinite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ countable_countable(A,A5)
     => ~ aa(set(A),$o,finite_finite2(A),A5) ) ).

% uncountable_infinite
tff(fact_7737_countable__finite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => countable_countable(A,S2) ) ).

% countable_finite
tff(fact_7738_uncountable__def,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ countable_countable(A,A5)
    <=> ( ( A5 != bot_bot(set(A)) )
        & ~ ? [F7: fun(nat,A)] : aa(set(nat),set(A),image2(nat,A,F7),top_top(set(nat))) = A5 ) ) ).

% uncountable_def
tff(fact_7739_countable__infiniteE_H,axiom,
    ! [A: $tType,A5: set(A)] :
      ( countable_countable(A,A5)
     => ( ~ aa(set(A),$o,finite_finite2(A),A5)
       => ~ ! [G2: fun(nat,A)] : ~ bij_betw(nat,A,G2,top_top(set(nat)),A5) ) ) ).

% countable_infiniteE'
tff(fact_7740_less__ccSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(B)
        & linorder(B) )
     => ! [A5: set(A),A2: B,F3: fun(A,B)] :
          ( countable_countable(A,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5)))
          <=> ? [X4: A] :
                ( member(A,X4,A5)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(A,B,F3,X4)) ) ) ) ) ).

% less_ccSUP_iff
tff(fact_7741_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_7742_ccSup__subset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( countable_countable(A,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ) ).

% ccSup_subset_mono
tff(fact_7743_butlast__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(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)) = $ite(Ys = nil(A),aa(list(A),list(A),butlast(A),Xs),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_7744_ccInf__less__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S2: set(A),A2: A] :
          ( countable_countable(A,S2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2)
          <=> ? [X4: A] :
                ( member(A,X4,S2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A2) ) ) ) ) ).

% ccInf_less_iff
tff(fact_7745_ccInf__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( countable_countable(A,B5)
         => ( countable_countable(A,A5)
           => ( ! [B4: A] :
                  ( member(A,B4,B5)
                 => ? [X: A] :
                      ( member(A,X,A5)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B4) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ) ) ).

% ccInf_mono
tff(fact_7746_ccInf__lower,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),Xb: A] :
          ( countable_countable(A,A5)
         => ( member(A,Xb,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),Xb) ) ) ) ).

% ccInf_lower
tff(fact_7747_ccInf__lower2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),U: A,V: A] :
          ( countable_countable(A,A5)
         => ( member(A,U,A5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),V) ) ) ) ) ).

% ccInf_lower2
tff(fact_7748_le__ccInf__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),B2: A] :
          ( countable_countable(A,A5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A5))
          <=> ! [X4: A] :
                ( member(A,X4,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X4) ) ) ) ) ).

% le_ccInf_iff
tff(fact_7749_ccInf__greatest,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),Z: A] :
          ( countable_countable(A,A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A5)) ) ) ) ).

% ccInf_greatest
tff(fact_7750_ccSup__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( countable_countable(A,B5)
         => ( countable_countable(A,A5)
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => ? [X: A] :
                      ( member(A,X,B5)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ) ) ).

% ccSup_mono
tff(fact_7751_ccSup__least,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),Z: A] :
          ( countable_countable(A,A5)
         => ( ! [X3: A] :
                ( member(A,X3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z) ) ) ) ).

% ccSup_least
tff(fact_7752_ccSup__upper,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),Xb: A] :
          ( countable_countable(A,A5)
         => ( member(A,Xb,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ).

% ccSup_upper
tff(fact_7753_ccSup__le__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),B2: A] :
          ( countable_countable(A,A5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B2)
          <=> ! [X4: A] :
                ( member(A,X4,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B2) ) ) ) ) ).

% ccSup_le_iff
tff(fact_7754_ccSup__upper2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),U: A,V: A] :
          ( countable_countable(A,A5)
         => ( member(A,U,A5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V),U)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ) ).

% ccSup_upper2
tff(fact_7755_less__ccSup__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S2: set(A),A2: A] :
          ( countable_countable(A,S2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2))
          <=> ? [X4: A] :
                ( member(A,X4,S2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4) ) ) ) ) ).

% less_ccSup_iff
tff(fact_7756_butlast_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(Xs = nil(A),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),butlast(A),Xs))) ).

% butlast.simps(2)
tff(fact_7757_butlast_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),butlast(A),nil(A)) = nil(A) ).

% butlast.simps(1)
tff(fact_7758_drop__butlast,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,Nb,aa(list(A),list(A),butlast(A),Xs)) = aa(list(A),list(A),butlast(A),drop(A,Nb,Xs)) ).

% drop_butlast
tff(fact_7759_in__set__butlastD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_butlastD
tff(fact_7760_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_7761_in__set__butlast__appendI,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] :
      ( ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs)))
        | member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Ys))) )
     => member(A,Xb,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_7762_ccInf__superset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( countable_countable(A,A5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ) ).

% ccInf_superset_mono
tff(fact_7763_map__butlast,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),butlast(B),Xs)) = aa(list(A),list(A),butlast(A),aa(list(B),list(A),map(B,A,F3),Xs)) ).

% map_butlast
tff(fact_7764_ccSUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( counta3822494911875563373attice(C)
     => ! [A5: set(A),B5: set(B),F3: fun(A,C),G3: fun(B,C)] :
          ( countable_countable(A,A5)
         => ( countable_countable(B,B5)
           => ( ! [N2: A] :
                  ( member(A,N2,A5)
                 => ? [X: B] :
                      ( member(B,X,B5)
                      & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,N2)),aa(B,C,G3,X)) ) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,F3),A5))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G3),B5))) ) ) ) ) ).

% ccSUP_mono
tff(fact_7765_ccSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),F3: fun(A,B),U: B] :
          ( countable_countable(A,A5)
         => ( ! [I3: A] :
                ( member(A,I3,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I3)),U) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),U) ) ) ) ).

% ccSUP_least
tff(fact_7766_ccSUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),I2: A,F3: fun(A,B)] :
          ( countable_countable(A,A5)
         => ( member(A,I2,A5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ).

% ccSUP_upper
tff(fact_7767_ccSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),F3: fun(A,B),U: B] :
          ( countable_countable(A,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),U)
          <=> ! [X4: A] :
                ( member(A,X4,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),U) ) ) ) ) ).

% ccSUP_le_iff
tff(fact_7768_ccSUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),I2: A,U: B,F3: fun(A,B)] :
          ( countable_countable(A,A5)
         => ( member(A,I2,A5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,I2))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ) ).

% ccSUP_upper2
tff(fact_7769_ccINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(B)
        & linorder(B) )
     => ! [A5: set(A),F3: fun(A,B),A2: B] :
          ( countable_countable(A,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),A2)
          <=> ? [X4: A] :
                ( member(A,X4,A5)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),A2) ) ) ) ) ).

% ccINF_less_iff
tff(fact_7770_ccINF__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( counta3822494911875563373attice(C)
     => ! [A5: set(A),B5: set(B),F3: fun(A,C),G3: fun(B,C)] :
          ( countable_countable(A,A5)
         => ( countable_countable(B,B5)
           => ( ! [M: B] :
                  ( member(B,M,B5)
                 => ? [X: A] :
                      ( member(A,X,A5)
                      & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,X)),aa(B,C,G3,M)) ) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,F3),A5))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G3),B5))) ) ) ) ) ).

% ccINF_mono
tff(fact_7771_ccINF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),I2: A,F3: fun(A,B)] :
          ( countable_countable(A,A5)
         => ( member(A,I2,A5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(A,B,F3,I2)) ) ) ) ).

% ccINF_lower
tff(fact_7772_ccINF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),I2: A,F3: fun(A,B),U: B] :
          ( countable_countable(A,A5)
         => ( member(A,I2,A5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),U)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),U) ) ) ) ) ).

% ccINF_lower2
tff(fact_7773_le__ccINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),U: B,F3: fun(A,B)] :
          ( countable_countable(A,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5)))
          <=> ! [X4: A] :
                ( member(A,X4,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,X4)) ) ) ) ) ).

% le_ccINF_iff
tff(fact_7774_ccINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),U: B,F3: fun(A,B)] :
          ( countable_countable(A,A5)
         => ( ! [I3: A] :
                ( member(A,I3,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,I3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ).

% ccINF_greatest
tff(fact_7775_to__nat__on__surj,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( countable_countable(A,A5)
     => ( ~ aa(set(A),$o,finite_finite2(A),A5)
       => ? [X3: A] :
            ( member(A,X3,A5)
            & ( aa(A,nat,countable_to_nat_on(A,A5),X3) = Nb ) ) ) ) ).

% to_nat_on_surj
tff(fact_7776_countableE__infinite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ~ aa(set(A),$o,finite_finite2(A),S2)
       => ~ ! [E2: fun(A,nat)] : ~ bij_betw(A,nat,E2,S2,top_top(set(nat))) ) ) ).

% countableE_infinite
tff(fact_7777_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_7778_nth__butlast,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),butlast(A),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),butlast(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% nth_butlast
tff(fact_7779_take__butlast,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,Nb,aa(list(A),list(A),butlast(A),Xs)) = take(A,Nb,Xs) ) ) ).

% take_butlast
tff(fact_7780_ccSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( countable_countable(A,A5)
         => ( countable_countable(A,B5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ) ).

% ccSup_inter_less_eq
tff(fact_7781_less__eq__ccInf__inter,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( countable_countable(A,A5)
         => ( countable_countable(A,B5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))),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)),A5),B5))) ) ) ) ).

% less_eq_ccInf_inter
tff(fact_7782_ccSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( countable_countable(A,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [X3: A] :
                  ( member(A,X3,A5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G3),B5))) ) ) ) ) ).

% ccSUP_subset_mono
tff(fact_7783_ccINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( countable_countable(A,A5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
           => ( ! [X3: A] :
                  ( member(A,X3,B5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G3),B5))) ) ) ) ) ).

% ccINF_superset_mono
tff(fact_7784_mono__ccSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => ( countable_countable(A,A5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F3),A5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ).

% mono_ccSup
tff(fact_7785_mono__ccSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),I6: set(C),A5: fun(C,A)] :
          ( order_mono(A,B,F3)
         => ( countable_countable(C,I6)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_akd(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5)),I6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A5),I6)))) ) ) ) ).

% mono_ccSUP
tff(fact_7786_mono__ccINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta4013691401010221786attice(A) )
     => ! [F3: fun(A,B),I6: set(C),A5: fun(C,A)] :
          ( order_mono(A,B,F3)
         => ( countable_countable(C,I6)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A5),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_akd(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5)),I6))) ) ) ) ).

% mono_ccINF
tff(fact_7787_mono__ccInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => ( countable_countable(A,A5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F3),A5))) ) ) ) ).

% mono_ccInf
tff(fact_7788_countable__as__injective__image,axiom,
    ! [A: $tType,A5: set(A)] :
      ( countable_countable(A,A5)
     => ( ~ aa(set(A),$o,finite_finite2(A),A5)
       => ~ ! [F2: fun(nat,A)] :
              ( ( A5 = aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat))) )
             => ~ inj_on(nat,A,F2,top_top(set(nat))) ) ) ) ).

% countable_as_injective_image
tff(fact_7789_image__to__nat__on,axiom,
    ! [A: $tType,A5: set(A)] :
      ( countable_countable(A,A5)
     => ( ~ aa(set(A),$o,finite_finite2(A),A5)
       => ( aa(set(A),set(nat),image2(A,nat,countable_to_nat_on(A,A5)),A5) = top_top(set(nat)) ) ) ) ).

% image_to_nat_on
tff(fact_7790_to__nat__on__infinite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ~ aa(set(A),$o,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_7791_butlast__conv__take,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xs) ).

% butlast_conv_take
tff(fact_7792_butlast__list__update,axiom,
    ! [A: $tType,Xs: list(A),K2: nat,Xb: A] :
      aa(list(A),list(A),butlast(A),list_update(A,Xs,K2,Xb)) = $ite(K2 = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),aa(list(A),list(A),butlast(A),Xs),list_update(A,aa(list(A),list(A),butlast(A),Xs),K2,Xb)) ).

% butlast_list_update
tff(fact_7793_countable__enum__cases,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ( aa(set(A),$o,finite_finite2(A),S2)
         => ! [F2: fun(A,nat)] : ~ bij_betw(A,nat,F2,S2,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),S2))) )
       => ~ ( ~ aa(set(A),$o,finite_finite2(A),S2)
           => ! [F2: fun(A,nat)] : ~ bij_betw(A,nat,F2,S2,top_top(set(nat))) ) ) ) ).

% countable_enum_cases
tff(fact_7794_butlast__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),butlast(A),take(A,Nb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_7795_sort__key__conv__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A)] :
          ( inj_on(A,B,F3,aa(list(A),set(A),set2(A),Xs))
         => ( linorder_sort_key(A,B,F3,Xs) = aa(list(A),list(A),fold(A,list(A),linorder_insort_key(A,B,F3),Xs),nil(A)) ) ) ) ).

% sort_key_conv_fold
tff(fact_7796_range__from__nat__into,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( countable_countable(A,A5)
       => ( aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A5)),top_top(set(nat))) = A5 ) ) ) ).

% range_from_nat_into
tff(fact_7797_sort__upt,axiom,
    ! [Mb: nat,Nb: nat] : linorder_sort_key(nat,nat,aTP_Lamp_ca(nat,nat),upt(Mb,Nb)) = upt(Mb,Nb) ).

% sort_upt
tff(fact_7798_sort__upto,axiom,
    ! [I2: int,J: int] : linorder_sort_key(int,int,aTP_Lamp_bl(int,int),upto(I2,J)) = upto(I2,J) ).

% sort_upto
tff(fact_7799_sort__key__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B)] : linorder_sort_key(A,B,F3,nil(A)) = nil(A) ) ).

% sort_key_simps(1)
tff(fact_7800_set__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A)] : aa(list(A),set(A),set2(A),linorder_sort_key(A,B,F3,Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_sort
tff(fact_7801_length__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),linorder_sort_key(A,B,F3,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ).

% length_sort
tff(fact_7802_distinct__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A)] :
          ( distinct(A,linorder_sort_key(A,B,F3,Xs))
        <=> distinct(A,Xs) ) ) ).

% distinct_sort
tff(fact_7803_from__nat__into__inject,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( countable_countable(A,A5)
       => ( ( B5 != bot_bot(set(A)) )
         => ( countable_countable(A,B5)
           => ( ( aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A5) = aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),B5) )
            <=> ( A5 = B5 ) ) ) ) ) ) ).

% from_nat_into_inject
tff(fact_7804_from__nat__into__inj__infinite,axiom,
    ! [A: $tType,A5: set(A),Mb: nat,Nb: nat] :
      ( countable_countable(A,A5)
     => ( ~ aa(set(A),$o,finite_finite2(A),A5)
       => ( ( aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A5),Mb) = aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A5),Nb) )
        <=> ( Mb = Nb ) ) ) ) ).

% from_nat_into_inj_infinite
tff(fact_7805_sort__key__simps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A,Xs: list(A)] : linorder_sort_key(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),linorder_sort_key(A,B,F3,Xs)) ) ).

% sort_key_simps(2)
tff(fact_7806_to__nat__on__from__nat__into__infinite,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( countable_countable(A,A5)
     => ( ~ aa(set(A),$o,finite_finite2(A),A5)
       => ( aa(A,nat,countable_to_nat_on(A,A5),aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A5),Nb)) = Nb ) ) ) ).

% to_nat_on_from_nat_into_infinite
tff(fact_7807_filter__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P2: fun(A,$o),F3: fun(A,B),Xs: list(A)] : aa(list(A),list(A),filter2(A,P2),linorder_sort_key(A,B,F3,Xs)) = linorder_sort_key(A,B,F3,aa(list(A),list(A),filter2(A,P2),Xs)) ) ).

% filter_sort
tff(fact_7808_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),K2: B,Xs: list(A)] : aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_ake(fun(A,B),fun(B,fun(A,$o)),F3),K2)),linorder_sort_key(A,B,F3,Xs)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_ake(fun(A,B),fun(B,fun(A,$o)),F3),K2)),Xs) ) ).

% sort_key_stable
tff(fact_7809_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C3: B,Xs: list(A)] : linorder_sort_key(A,B,aTP_Lamp_qj(B,fun(A,B),C3),Xs) = Xs ) ).

% sort_key_const
tff(fact_7810_from__nat__into,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( ( A5 != bot_bot(set(A)) )
     => member(A,aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A5),Nb),A5) ) ).

% from_nat_into
tff(fact_7811_inj__on__from__nat__into,axiom,
    ! [A: $tType] : inj_on(set(A),fun(nat,A),counta4804993851260445106t_into(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_akf(set(A),$o))) ).

% inj_on_from_nat_into
tff(fact_7812_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( linorder_sort_key(A,A,aTP_Lamp_mr(A,A),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_7813_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),linorder_sort_key(A,A,aTP_Lamp_mr(A,A),Xs)) ) ).

% sorted_sort
tff(fact_7814_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),linorder_sort_key(B,A,F3,Xs))) ) ).

% sorted_sort_key
tff(fact_7815_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)) = linorder_sort_key(A,A,aTP_Lamp_mr(A,A),remdups(A,Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_7816_range__from__nat__into__subset,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( A5 != bot_bot(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A5)),top_top(set(nat)))),A5) ) ).

% range_from_nat_into_subset
tff(fact_7817_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : linorder_sort_key(A,A,aTP_Lamp_mr(A,A),Xs) = aa(list(A),list(A),fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_mr(A,A)),Xs),nil(A)) ) ).

% sort_conv_fold
tff(fact_7818_bij__betw__from__nat__into__finite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => bij_betw(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),S2),aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),S2)),S2) ) ).

% bij_betw_from_nat_into_finite
tff(fact_7819_bij__betw__from__nat__into,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ~ aa(set(A),$o,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_7820_sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A)] : linorder_sort_key(A,B,F3,Xs) = aa(list(A),list(A),foldr(A,list(A),linorder_insort_key(A,B,F3),Xs),nil(A)) ) ).

% sort_key_def
tff(fact_7821_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,$o)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P2) = aa(list(A),A,case_list(A,A,abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P2),aTP_Lamp_akg(A,fun(list(A),A))),aa(list(A),list(A),filter2(A,P2),linorder_sort_key(A,A,aTP_Lamp_mr(A,A),Xs))) ) ).

% Bleast_code
tff(fact_7822_inter__coset__fold,axiom,
    ! [A: $tType,A5: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),coset(A,Xs)) = aa(set(A),set(A),fold(A,set(A),remove(A),Xs),A5) ).

% inter_coset_fold
tff(fact_7823_UNIV__coset,axiom,
    ! [A: $tType] : top_top(set(A)) = coset(A,nil(A)) ).

% UNIV_coset
tff(fact_7824_subset__code_I2_J,axiom,
    ! [A: $tType,A5: set(A),Ys: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),coset(A,Ys))
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
         => ~ member(A,X4,A5) ) ) ).

% subset_code(2)
tff(fact_7825_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_7826_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_7827_insert__code_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),coset(A,Xs)) = coset(A,aa(list(A),list(A),removeAll(A,Xb),Xs)) ).

% insert_code(2)
tff(fact_7828_union__coset__filter,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xs)),A5) = coset(A,aa(list(A),list(A),filter2(A,aTP_Lamp_akh(set(A),fun(A,$o),A5)),Xs)) ).

% union_coset_filter
tff(fact_7829_subset__code_I3_J,axiom,
    ! [A: $tType] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),coset(A,nil(A))),aa(list(A),set(A),set2(A),nil(A))) ).

% subset_code(3)
tff(fact_7830_minus__coset__filter,axiom,
    ! [A: $tType,A5: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),coset(A,Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_a(set(A),fun(A,$o),A5)),Xs)) ).

% minus_coset_filter
tff(fact_7831_comp__fun__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
    <=> ( finite4664212375090638736ute_on(A,B,S2,F3)
        & finite4980608107308702382axioms(A,B,S2,F3) ) ) ).

% comp_fun_idem_on_def
tff(fact_7832_comp__fun__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( finite4980608107308702382axioms(A,B,S2,F3)
       => finite673082921795544331dem_on(A,B,S2,F3) ) ) ).

% comp_fun_idem_on.intro
tff(fact_7833_comp__fun__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A] :
          ( member(A,X3,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,X3)) = aa(A,fun(B,B),F3,X3) ) )
     => finite4980608107308702382axioms(A,B,S2,F3) ) ).

% comp_fun_idem_on_axioms.intro
tff(fact_7834_comp__fun__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite4980608107308702382axioms(A,B,S2,F3)
    <=> ! [X4: A] :
          ( member(A,X4,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X4)),aa(A,fun(B,B),F3,X4)) = aa(A,fun(B,B),F3,X4) ) ) ) ).

% comp_fun_idem_on_axioms_def
tff(fact_7835_comp__fun__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => finite4980608107308702382axioms(A,B,S2,F3) ) ).

% comp_fun_idem_on.axioms(2)
tff(fact_7836_shuffles_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Yb: set(list(A))] :
      ( ( shuffles(A,Xb,Xaa) = Yb )
     => ( 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)),Xb),Xaa))
       => ( ( ( Xb = nil(A) )
           => ( ( Yb = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xaa),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)),Xaa)) ) )
         => ( ( ( Xaa = nil(A) )
             => ( ( Yb = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xb),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)),Xb),nil(A))) ) )
           => ~ ! [X3: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ! [Y: A,Ys3: list(A)] :
                      ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3) )
                     => ( ( Yb = 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),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)))),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),X3),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),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_7837_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))),insert(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)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),field2(A,R2)) ).

% Field_insert
tff(fact_7838_Field__empty,axiom,
    ! [A: $tType] : field2(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ).

% Field_empty
tff(fact_7839_underS__Field3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( ( field2(A,R2) != bot_bot(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A2)),field2(A,R2)) ) ).

% underS_Field3
tff(fact_7840_underS__empty,axiom,
    ! [A: $tType,A2: A,R2: set(product_prod(A,A))] :
      ( ~ member(A,A2,field2(A,R2))
     => ( order_underS(A,R2,A2) = bot_bot(set(A)) ) ) ).

% underS_empty
tff(fact_7841_underS__Field2,axiom,
    ! [A: $tType,A2: A,R2: set(product_prod(A,A))] :
      ( member(A,A2,field2(A,R2))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A2)),field2(A,R2)) ) ).

% underS_Field2
tff(fact_7842_finite__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
     => aa(set(A),$o,finite_finite2(A),field2(A,R2)) ) ).

% finite_Field
tff(fact_7843_FieldI2,axiom,
    ! [A: $tType,I2: A,J: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J),R)
     => member(A,J,field2(A,R)) ) ).

% FieldI2
tff(fact_7844_FieldI1,axiom,
    ! [A: $tType,I2: A,J: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I2),J),R)
     => member(A,I2,field2(A,R)) ) ).

% FieldI1
tff(fact_7845_shuffles_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P2: fun(list(A),fun(list(A),$o))] :
      ( 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))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,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)))
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,Xs2),nil(A)) )
         => ( ! [X3: A,Xs2: list(A),Y: 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),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)))
               => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3))
                 => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3)
                   => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)) ) ) )
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,A0),A1) ) ) ) ) ).

% shuffles.pinduct
tff(fact_7846_Field__natLeq__on,axiom,
    ! [Nb: nat] : field2(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ajj(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb)) ).

% Field_natLeq_on
tff(fact_7847_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))),insert(list(A)),Ys),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_7848_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))),insert(list(A)),Xs),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_7849_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Yb: 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),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)))
     => ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),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),Xb)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys))) ) ) ).

% shuffles.psimps(3)
tff(fact_7850_underS__incl__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( member(A,A2,field2(A,R2))
       => ( member(A,B2,field2(A,R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A2)),order_underS(A,R2,B2))
          <=> 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_7851_Under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : order_Under(A,R2,A5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aki(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A5)) ).

% Under_def
tff(fact_7852_UnderS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : order_UnderS(A,R2,A5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_akj(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A5)) ).

% UnderS_def
tff(fact_7853_Above__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : order_Above(A,R2,A5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_akk(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A5)) ).

% Above_def
tff(fact_7854_finite__Linear__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: A,P2: fun(A,$o)] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( member(A,Xb,field2(A,R2))
       => ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
         => ( ! [X3: A] :
                ( member(A,X3,field2(A,R2))
               => ( ! [Y3: A] :
                      ( member(A,Y3,order_aboveS(A,R2,X3))
                     => aa(A,$o,P2,Y3) )
                 => aa(A,$o,P2,X3) ) )
           => aa(A,$o,P2,Xb) ) ) ) ) ).

% finite_Linear_order_induct
tff(fact_7855_aboveS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : order_aboveS(A,R2,A2) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_akl(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A2)) ).

% aboveS_def
tff(fact_7856_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P2: fun(list(A),fun(list(A),$o))] :
      ( 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))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,nil(A)),Ys3) )
       => ( ! [X3: 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),X3),Xs2)),Ys3))
             => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,Ys3),Xs2)
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3) ) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,A0),A1) ) ) ) ).

% splice.pinduct
tff(fact_7857_cofinal__def,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,A5,R2)
    <=> ! [X4: A] :
          ( member(A,X4,field2(A,R2))
         => ? [Xa2: A] :
              ( member(A,Xa2,A5)
              & ( X4 != Xa2 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R2) ) ) ) ).

% cofinal_def
tff(fact_7858_splice_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Yb: list(A)] :
      ( ( splice(A,Xb,Xaa) = Yb )
     => ( 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)),Xb),Xaa))
       => ( ( ( Xb = nil(A) )
           => ( ( Yb = Xaa )
             => ~ 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)),Xaa)) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( ( Yb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),splice(A,Xaa,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),X3),Xs2)),Xaa)) ) ) ) ) ) ).

% splice.pelims
tff(fact_7859_Refl__under__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( refl_on(A,field2(A,R2),R2)
     => ( 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)),insert(A),A2),bot_bot(set(A)))) ) ) ) ).

% Refl_under_underS
tff(fact_7860_split__Nil__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( splice(A,Xs,Ys) = nil(A) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% split_Nil_iff
tff(fact_7861_splice__Nil2,axiom,
    ! [A: $tType,Xs: list(A)] : splice(A,Xs,nil(A)) = Xs ).

% splice_Nil2
tff(fact_7862_splice__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : member(list(A),splice(A,Xs,Ys),shuffles(A,Xs,Ys)) ).

% splice_in_shuffles
tff(fact_7863_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),splice(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_splice
tff(fact_7864_splice__replicate,axiom,
    ! [A: $tType,Mb: nat,Xb: A,Nb: nat] : splice(A,replicate(A,Mb,Xb),replicate(A,Nb,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb),Xb) ).

% splice_replicate
tff(fact_7865_under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : order_under(A,R2,A2) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_akm(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),A2)) ).

% under_def
tff(fact_7866_splice_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Yb: list(A)] :
      ( ( splice(A,Xb,Xaa) = Yb )
     => ( ( ( Xb = nil(A) )
         => ( Yb != Xaa ) )
       => ~ ! [X3: A,Xs2: list(A)] :
              ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ( Yb != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),splice(A,Xaa,Xs2)) ) ) ) ) ).

% splice.elims
tff(fact_7867_splice_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] : splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),splice(A,Ys,Xs)) ).

% splice.simps(2)
tff(fact_7868_splice_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : splice(A,nil(A),Ys) = Ys ).

% splice.simps(1)
tff(fact_7869_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,Xb: 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),Xb),Xs)),Ys))
     => ( splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),splice(A,Ys,Xs)) ) ) ).

% splice.psimps(2)
tff(fact_7870_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))
     => ( splice(A,nil(A),Ys) = Ys ) ) ).

% splice.psimps(1)
tff(fact_7871_Total__subset__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R2),R2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
       => ( ( R2 = bot_bot(set(product_prod(A,A))) )
          | ? [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))),insert(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_7872_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z: B,Yb: B,A2: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),Yb)
         => ( member(A,A2,A5)
           => ? [Y8: B] :
                ( ( Yb = aa(B,B,aa(A,fun(B,B),F3,A2),Y8) )
                & aa(B,$o,finite_fold_graph(A,B,F3,Z,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)),insert(A),A2),bot_bot(set(A))))),Y8) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_7873_IdI,axiom,
    ! [A: $tType,A2: A] : 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),id2(A)) ).

% IdI
tff(fact_7874_pair__in__Id__conv,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),id2(A))
    <=> ( A2 = B2 ) ) ).

% pair_in_Id_conv
tff(fact_7875_rtrancl__empty,axiom,
    ! [A: $tType] : transitive_rtrancl(A,bot_bot(set(product_prod(A,A)))) = id2(A) ).

% rtrancl_empty
tff(fact_7876_pair__leq__def,axiom,
    fun_pair_leq = aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),sup_sup(set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),fun_pair_less),id2(product_prod(nat,nat))) ).

% pair_leq_def
tff(fact_7877_relpow_Osimps_I1_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R) = id2(A) ).

% relpow.simps(1)
tff(fact_7878_Id__def,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_akn(product_prod(A,A),$o)) ).

% Id_def
tff(fact_7879_IdD,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),id2(A))
     => ( A2 = B2 ) ) ).

% IdD
tff(fact_7880_IdE,axiom,
    ! [A: $tType,P: product_prod(A,A)] :
      ( member(product_prod(A,A),P,id2(A))
     => ~ ! [X3: A] : P != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).

% IdE
tff(fact_7881_fold__graph_OemptyI,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z: B] : aa(B,$o,finite_fold_graph(A,B,F3,Z,bot_bot(set(A))),Z) ).

% fold_graph.emptyI
tff(fact_7882_empty__fold__graphE,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z: B,Xb: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F3,Z,bot_bot(set(A))),Xb)
     => ( Xb = Z ) ) ).

% empty_fold_graphE
tff(fact_7883_comp__fun__commute__on_Ofold__graph__finite,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Z: B,A5: set(A),Yb: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),Yb)
       => aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% comp_fun_commute_on.fold_graph_finite
tff(fact_7884_comp__fun__commute__on_Ofold__graph__determ,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z: B,Xb: B,Yb: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),Xb)
         => ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),Yb)
           => ( Yb = Xb ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_determ
tff(fact_7885_finite__imp__fold__graph,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,fun(B,B)),Z: B] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [X_12: B] : aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),X_12) ) ).

% finite_imp_fold_graph
tff(fact_7886_fold__graph__closed__lemma,axiom,
    ! [A: $tType,B: $tType,G3: fun(A,fun(B,B)),Z: B,A5: set(A),Xb: B,B5: set(B),F3: fun(A,fun(B,B))] :
      ( aa(B,$o,finite_fold_graph(A,B,G3,Z,A5),Xb)
     => ( ! [A4: A,B4: B] :
            ( member(A,A4,A5)
           => ( member(B,B4,B5)
             => ( aa(B,B,aa(A,fun(B,B),F3,A4),B4) = aa(B,B,aa(A,fun(B,B),G3,A4),B4) ) ) )
       => ( ! [A4: A,B4: B] :
              ( member(A,A4,A5)
             => ( member(B,B4,B5)
               => member(B,aa(B,B,aa(A,fun(B,B),G3,A4),B4),B5) ) )
         => ( member(B,Z,B5)
           => ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),Xb)
              & member(B,Xb,B5) ) ) ) ) ) ).

% fold_graph_closed_lemma
tff(fact_7887_fold__graph__closed__eq,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B),F3: fun(A,fun(B,B)),G3: fun(A,fun(B,B)),Z: B] :
      ( ! [A4: A,B4: B] :
          ( member(A,A4,A5)
         => ( member(B,B4,B5)
           => ( aa(B,B,aa(A,fun(B,B),F3,A4),B4) = aa(B,B,aa(A,fun(B,B),G3,A4),B4) ) ) )
     => ( ! [A4: A,B4: B] :
            ( member(A,A4,A5)
           => ( member(B,B4,B5)
             => member(B,aa(B,B,aa(A,fun(B,B),G3,A4),B4),B5) ) )
       => ( member(B,Z,B5)
         => ( finite_fold_graph(A,B,F3,Z,A5) = finite_fold_graph(A,B,G3,Z,A5) ) ) ) ) ).

% fold_graph_closed_eq
tff(fact_7888_fold__graph_Ocases,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z: B,A1: set(A),A22: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A1),A22)
     => ( ( ( A1 = bot_bot(set(A)) )
         => ( A22 != Z ) )
       => ~ ! [X3: A,A6: set(A)] :
              ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),A6) )
             => ! [Y: B] :
                  ( ( A22 = aa(B,B,aa(A,fun(B,B),F3,X3),Y) )
                 => ( ~ member(A,X3,A6)
                   => ~ aa(B,$o,finite_fold_graph(A,B,F3,Z,A6),Y) ) ) ) ) ) ).

% fold_graph.cases
tff(fact_7889_fold__graph_Osimps,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B)),Z: B,A1: set(A),A22: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A1),A22)
    <=> ( ( ( A1 = bot_bot(set(A)) )
          & ( A22 = Z ) )
        | ? [X4: A,A8: set(A),Y4: B] :
            ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),A8) )
            & ( A22 = aa(B,B,aa(A,fun(B,B),F3,X4),Y4) )
            & ~ member(A,X4,A8)
            & aa(B,$o,finite_fold_graph(A,B,F3,Z,A8),Y4) ) ) ) ).

% fold_graph.simps
tff(fact_7890_fold__graph_OinsertI,axiom,
    ! [A: $tType,B: $tType,Xb: A,A5: set(A),F3: fun(A,fun(B,B)),Z: B,Yb: B] :
      ( ~ member(A,Xb,A5)
     => ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),Yb)
       => aa(B,$o,finite_fold_graph(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),aa(B,B,aa(A,fun(B,B),F3,Xb),Yb)) ) ) ).

% fold_graph.insertI
tff(fact_7891_fold__graph__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: fun(A,B),A5: set(A),F3: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G3,A5)
     => ( finite_fold_graph(B,C,F3,Z,aa(set(A),set(B),image2(A,B,G3),A5)) = finite_fold_graph(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F3),G3),Z,A5) ) ) ).

% fold_graph_image
tff(fact_7892_comp__fun__commute__on_Ofold__graph__insertE,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B,V: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),S2)
       => ( aa(B,$o,finite_fold_graph(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),A5)),V)
         => ( ~ member(A,Xb,A5)
           => ~ ! [Y: B] :
                  ( ( V = aa(B,B,aa(A,fun(B,B),F3,Xb),Y) )
                 => ~ aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),Y) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE
tff(fact_7893_comp__fun__commute__on_Ofold__equality,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z: B,Yb: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),Yb)
         => ( finite_fold(A,B,F3,Z,A5) = Yb ) ) ) ) ).

% comp_fun_commute_on.fold_equality
tff(fact_7894_reflcl__set__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Xa: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),aTP_Lamp_yb(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),fequal(A)),X),Xa)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),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),id2(A))) ) ).

% reflcl_set_eq
tff(fact_7895_Finite__Set_Ofold__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),Z: A,A5: set(B)] :
      finite_fold(B,A,F3,Z,A5) = $ite(aa(set(B),$o,finite_finite2(B),A5),the(A,finite_fold_graph(B,A,F3,Z,A5)),Z) ).

% Finite_Set.fold_def
tff(fact_7896_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)
     => ( member(A,A2,field2(A,R2))
       => ( member(A,B2,field2(A,R2))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
          <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),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),id2(A))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_7897_comp__fun__commute__on_Ofold__graph__fold,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => aa(B,$o,finite_fold_graph(A,B,F3,Z,A5),finite_fold(A,B,F3,Z,A5)) ) ) ) ).

% comp_fun_commute_on.fold_graph_fold
tff(fact_7898_bsqr__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R2) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_akp(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),R2)))) ).

% bsqr_def
tff(fact_7899_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
      <=> ! [A8: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),field2(A,R2))
           => ( ( A8 != bot_bot(set(A)) )
             => ? [X4: A] :
                  ( member(A,X4,A8)
                  & ! [Xa2: A] :
                      ( member(A,Xa2,A8)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_7900_wf__empty,axiom,
    ! [A: $tType] : wf(A,bot_bot(set(product_prod(A,A)))) ).

% wf_empty
tff(fact_7901_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_7902_wf__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lex(A,R2)) ) ).

% wf_lex
tff(fact_7903_wf__lenlex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lenlex(A,R2)) ) ).

% wf_lenlex
tff(fact_7904_wf__insert,axiom,
    ! [A: $tType,Yb: A,Xb: 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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Yb),Xb)),R2))
    <=> ( wf(A,R2)
        & ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),transitive_rtrancl(A,R2)) ) ) ).

% wf_insert
tff(fact_7905_reduction__pairI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,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_7906_wfI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A),B5: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_aie(set(A),fun(A,set(A)),B5)))
     => ( ! [X3: A,P6: fun(A,$o)] :
            ( ! [Xa: A] :
                ( ! [Y: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa),R2)
                   => aa(A,$o,P6,Y) )
               => aa(A,$o,P6,Xa) )
           => ( member(A,X3,A5)
             => ( member(A,X3,B5)
               => aa(A,$o,P6,X3) ) ) )
       => wf(A,R2) ) ) ).

% wfI
tff(fact_7907_wf__if__measure,axiom,
    ! [A: $tType,P2: fun(A,$o),F3: fun(A,nat),G3: fun(A,A)] :
      ( ! [X3: A] :
          ( aa(A,$o,P2,X3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,aa(A,A,G3,X3))),aa(A,nat,F3,X3)) )
     => wf(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_akq(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),P2),G3)))) ) ).

% wf_if_measure
tff(fact_7908_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),ord_less(A)))) ) ).

% wf
tff(fact_7909_wf__less,axiom,
    wf(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat)))) ).

% wf_less
tff(fact_7910_wf__lexn,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Nb: nat] :
      ( wf(A,R2)
     => wf(list(A),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),Nb)) ) ).

% wf_lexn
tff(fact_7911_wf__pair__less,axiom,
    wf(product_prod(nat,nat),fun_pair_less) ).

% wf_pair_less
tff(fact_7912_wf__int__ge__less__than2,axiom,
    ! [D3: int] : wf(int,int_ge_less_than2(D3)) ).

% wf_int_ge_less_than2
tff(fact_7913_wf__int__ge__less__than,axiom,
    ! [D3: int] : wf(int,int_ge_less_than(D3)) ).

% wf_int_ge_less_than
tff(fact_7914_wf__no__loop,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,R,R) = bot_bot(set(product_prod(A,A))) )
     => wf(A,R) ) ).

% wf_no_loop
tff(fact_7915_wfE__min_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z2: A] :
              ( member(A,Z2,Q)
             => ~ ! [Y3: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2),R)
                   => ~ member(A,Y3,Q) ) ) ) ) ).

% wfE_min'
tff(fact_7916_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ~ ? [F7: fun(nat,A)] :
          ! [I4: nat] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F7,aa(nat,nat,suc,I4))),aa(nat,A,F7,I4)),R2) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_7917_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),F3: fun(nat,A)] :
      ( wf(A,R2)
     => ~ ! [K: nat] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F3,aa(nat,nat,suc,K))),aa(nat,A,F3,K)),R2) ) ).

% wf_no_infinite_down_chainE
tff(fact_7918_wf__induct__rule,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P2: fun(A,$o),A2: A] :
      ( wf(A,R2)
     => ( ! [X3: A] :
            ( ! [Y3: A] :
                ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3),R2)
               => aa(A,$o,P2,Y3) )
           => aa(A,$o,P2,X3) )
       => aa(A,$o,P2,A2) ) ) ).

% wf_induct_rule
tff(fact_7919_wf__eq__minimal,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [Q7: set(A)] :
          ( ? [X4: A] : member(A,X4,Q7)
         => ? [X4: A] :
              ( member(A,X4,Q7)
              & ! [Y4: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X4),R2)
                 => ~ member(A,Y4,Q7) ) ) ) ) ).

% wf_eq_minimal
tff(fact_7920_wf__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( wf(A,R2)
     => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2),R2) ) ).

% wf_not_refl
tff(fact_7921_wf__not__sym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,Xb: A] :
      ( wf(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Xb),R2)
       => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),A2),R2) ) ) ).

% wf_not_sym
tff(fact_7922_wf__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( wf(A,R2)
     => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2),R2) ) ).

% wf_irrefl
tff(fact_7923_wf__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P2: fun(A,$o),A2: A] :
      ( wf(A,R2)
     => ( ! [X3: A] :
            ( ! [Y3: A] :
                ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3),R2)
               => aa(A,$o,P2,Y3) )
           => aa(A,$o,P2,X3) )
       => aa(A,$o,P2,A2) ) ) ).

% wf_induct
tff(fact_7924_wf__asym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,Xb: A] :
      ( wf(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Xb),R2)
       => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),A2),R2) ) ) ).

% wf_asym
tff(fact_7925_wfUNIVI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [P6: fun(A,$o),X3: A] :
          ( ! [Xa: A] :
              ( ! [Y: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa),R2)
                 => aa(A,$o,P6,Y) )
             => aa(A,$o,P6,Xa) )
         => aa(A,$o,P6,X3) )
     => wf(A,R2) ) ).

% wfUNIVI
tff(fact_7926_wfI__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X3: A,Q8: set(A)] :
          ( member(A,X3,Q8)
         => ? [Xa: A] :
              ( member(A,Xa,Q8)
              & ! [Y: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa),R)
                 => ~ member(A,Y,Q8) ) ) )
     => wf(A,R) ) ).

% wfI_min
tff(fact_7927_wfE__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xb: A,Q: set(A)] :
      ( wf(A,R)
     => ( member(A,Xb,Q)
       => ~ ! [Z2: A] :
              ( member(A,Z2,Q)
             => ~ ! [Y3: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z2),R)
                   => ~ member(A,Y3,Q) ) ) ) ) ).

% wfE_min
tff(fact_7928_wf__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [P5: fun(A,$o)] :
          ( ! [X4: A] :
              ( ! [Y4: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X4),R2)
                 => aa(A,$o,P5,Y4) )
             => aa(A,$o,P5,X4) )
         => ! [X_13: A] : aa(A,$o,P5,X_13) ) ) ).

% wf_def
tff(fact_7929_wf__bounded__measure,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F3: fun(A,nat)] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A4))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,B4)),aa(A,nat,Ub,A4))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,A4)),aa(A,nat,F3,B4)) ) )
     => wf(A,R2) ) ).

% wf_bounded_measure
tff(fact_7930_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P2: fun(B,$o),K2: B,Mb: fun(B,A)] :
      ( wf(A,R2)
     => ( ! [X3: A,Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),transitive_trancl(A,R2))
          <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X3),transitive_rtrancl(A,R2)) )
       => ( aa(B,$o,P2,K2)
         => ? [X3: B] :
              ( aa(B,$o,P2,X3)
              & ! [Y3: B] :
                  ( aa(B,$o,P2,Y3)
                 => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Mb,X3)),aa(B,A,Mb,Y3)),transitive_rtrancl(A,R2)) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_7931_reduction__pair__def,axiom,
    ! [A: $tType,P2: product_prod(set(product_prod(A,A)),set(product_prod(A,A)))] :
      ( fun_reduction_pair(A,P2)
    <=> ( wf(A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2))
        & aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P2))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2)) ) ) ).

% reduction_pair_def
tff(fact_7932_reduction__pair__lemma,axiom,
    ! [A: $tType,P2: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( fun_reduction_pair(A,P2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2))
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S2),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P2))
         => ( wf(A,S2)
           => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),S2)) ) ) ) ) ).

% reduction_pair_lemma
tff(fact_7933_wf__eq__minimal2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [A8: set(A)] :
          ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),field2(A,R2))
            & ( A8 != bot_bot(set(A)) ) )
         => ? [X4: A] :
              ( member(A,X4,A8)
              & ! [Xa2: A] :
                  ( member(A,Xa2,A8)
                 => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_7934_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,set(B)),F3: fun(A,set(B))] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R2)
         => ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Ub,A4))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Ub,B4)),aa(A,set(B),Ub,A4))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F3,B4)),aa(A,set(B),Ub,A4))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(A,set(B),F3,A4)),aa(A,set(B),F3,B4)) ) )
     => wf(A,R2) ) ).

% wf_bounded_set
tff(fact_7935_finite__subset__wf,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_akr(set(A),fun(set(A),fun(set(A),$o)),A5)))) ) ).

% finite_subset_wf
tff(fact_7936_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),P2: fun(fun(A,B),fun(A,fun(B,$o)))] :
      ( wf(A,R)
     => ( ! [F2: fun(A,B),G2: fun(A,B),X3: A,R3: B] :
            ( ! [Z3: A] :
                ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X3),R)
               => ( aa(A,B,F2,Z3) = aa(A,B,G2,Z3) ) )
           => ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P2,F2),X3),R3)
            <=> aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P2,G2),X3),R3) ) )
       => ( ! [X3: A,F2: fun(A,B)] :
              ( ! [Y3: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X3),R)
                 => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P2,F2),Y3),aa(A,B,F2,Y3)) )
             => ? [X_1: B] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P2,F2),X3),X_1) )
         => ? [F2: fun(A,B)] :
            ! [X: A] : aa(B,$o,aa(A,fun(B,$o),aa(fun(A,B),fun(A,fun(B,$o)),P2,F2),X),aa(A,B,F2,X)) ) ) ) ).

% dependent_wf_choice
tff(fact_7937_partial__order__on__well__order__on,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
     => ( order_7125193373082350890der_on(A,A5,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),id2(A))) ) ) ).

% partial_order_on_well_order_on
tff(fact_7938_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_7939_dependent__wellorder__choice,axiom,
    ! [A: $tType,B: $tType] :
      ( wellorder(B)
     => ! [P2: fun(fun(B,A),fun(B,fun(A,$o)))] :
          ( ! [R3: A,F2: fun(B,A),G2: fun(B,A),X3: B] :
              ( ! [Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y3),X3)
                 => ( aa(B,A,F2,Y3) = aa(B,A,G2,Y3) ) )
             => ( aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P2,F2),X3),R3)
              <=> aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P2,G2),X3),R3) ) )
         => ( ! [X3: B,F2: fun(B,A)] :
                ( ! [Y3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y3),X3)
                   => aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P2,F2),Y3),aa(B,A,F2,Y3)) )
               => ? [X_1: A] : aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P2,F2),X3),X_1) )
           => ? [F2: fun(B,A)] :
              ! [X: B] : aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P2,F2),X),aa(B,A,F2,X)) ) ) ) ).

% dependent_wellorder_choice
tff(fact_7940_finite__Partial__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: A,P2: fun(A,$o)] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => ( member(A,Xb,field2(A,R2))
       => ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
         => ( ! [X3: A] :
                ( member(A,X3,field2(A,R2))
               => ( ! [Y3: A] :
                      ( member(A,Y3,order_aboveS(A,R2,X3))
                     => aa(A,$o,P2,Y3) )
                 => aa(A,$o,P2,X3) ) )
           => aa(A,$o,P2,Xb) ) ) ) ) ).

% finite_Partial_order_induct
tff(fact_7941_chains__extend,axiom,
    ! [A: $tType,C3: set(set(A)),S2: set(set(A)),Z: set(A)] :
      ( member(set(set(A)),C3,chains2(A,S2))
     => ( member(set(A),Z,S2)
       => ( ! [X3: set(A)] :
              ( member(set(A),X3,C3)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Z) )
         => member(set(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Z),bot_bot(set(set(A))))),C3),chains2(A,S2)) ) ) ) ).

% chains_extend
tff(fact_7942_Zorns__po__lemma,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => ( ! [C6: set(A)] :
            ( member(set(A),C6,chains(A,R2))
           => ? [X: A] :
                ( member(A,X,field2(A,R2))
                & ! [Xa3: A] :
                    ( member(A,Xa3,C6)
                   => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X),R2) ) ) )
       => ? [X3: A] :
            ( member(A,X3,field2(A,R2))
            & ! [Xa: A] :
                ( member(A,Xa,field2(A,R2))
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa),R2)
                 => ( Xa = X3 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_7943_Chains__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : chains(A,R2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aks(set(product_prod(A,A)),fun(set(A),$o),R2)) ).

% Chains_def
tff(fact_7944_Chains__subset_H,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_yb(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))),chains(A,R2)) ) ).

% Chains_subset'
tff(fact_7945_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A2: list(A),B2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),A2),B2)
    <=> ? [Z5: list(product_prod(A,B))] :
          ( member(list(product_prod(A,B)),Z5,aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_akt(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)))
          & ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Z5) = A2 )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Z5) = B2 ) ) ) ).

% list.in_rel
tff(fact_7946_list__all2__Nil,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),nil(A)),Ys)
    <=> ( Ys = nil(B) ) ) ).

% list_all2_Nil
tff(fact_7947_list__all2__Nil2,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o)),Xs: list(A)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),nil(B))
    <=> ( Xs = nil(A) ) ) ).

% list_all2_Nil2
tff(fact_7948_list__all2__rev,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),aa(list(A),list(A),rev(A),Xs)),aa(list(B),list(B),rev(B),Ys))
    <=> aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys) ) ).

% list_all2_rev
tff(fact_7949_chains__alt__def,axiom,
    ! [A: $tType,A5: set(set(A))] : chains2(A,A5) = aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),pred_chain(set(A),A5,ord_less(set(A)))) ).

% chains_alt_def
tff(fact_7950_list_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,Sa: fun(A,fun(B,$o)),Xb: list(A),G3: fun(C,B),Yb: list(C)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Sa),Xb),aa(list(C),list(B),map(C,B,G3),Yb))
    <=> aa(list(C),$o,aa(list(A),fun(list(C),$o),list_all2(A,C,aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aku(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Sa),G3)),Xb),Yb) ) ).

% list.rel_map(2)
tff(fact_7951_list_Orel__map_I1_J,axiom,
    ! [C: $tType,A: $tType,B: $tType,Sb: fun(A,fun(B,$o)),I2: fun(C,A),Xb: list(C),Yb: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Sb),aa(list(C),list(A),map(C,A,I2),Xb)),Yb)
    <=> aa(list(B),$o,aa(list(C),fun(list(B),$o),list_all2(C,B,aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_akv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Sb),I2)),Xb),Yb) ) ).

% list.rel_map(1)
tff(fact_7952_list__all2__map1,axiom,
    ! [C: $tType,A: $tType,B: $tType,P2: fun(A,fun(B,$o)),F3: fun(C,A),As: list(C),Bs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),aa(list(C),list(A),map(C,A,F3),As)),Bs)
    <=> aa(list(B),$o,aa(list(C),fun(list(B),$o),list_all2(C,B,aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_akv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),P2),F3)),As),Bs) ) ).

% list_all2_map1
tff(fact_7953_list__all2__map2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: fun(A,fun(B,$o)),As: list(A),F3: fun(C,B),Bs: list(C)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),As),aa(list(C),list(B),map(C,B,F3),Bs))
    <=> aa(list(C),$o,aa(list(A),fun(list(C),$o),list_all2(A,C,aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aku(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),P2),F3)),As),Bs) ) ).

% list_all2_map2
tff(fact_7954_list__all2__rev1,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),aa(list(A),list(A),rev(A),Xs)),Ys)
    <=> aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),aa(list(B),list(B),rev(B),Ys)) ) ).

% list_all2_rev1
tff(fact_7955_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(B,$o,aa(A,fun(B,$o),P2,aa(nat,A,nth(A,Xs),I4)),aa(nat,B,nth(B,Ys),I4)) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_7956_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),P2: fun(A,fun(B,$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),A2))
           => aa(B,$o,aa(A,fun(B,$o),P2,aa(nat,A,nth(A,A2),N2)),aa(nat,B,nth(B,B2),N2)) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),A2),B2) ) ) ).

% list_all2_all_nthI
tff(fact_7957_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),P: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P),aa(list(B),nat,size_size(list(B)),Ys))
       => aa(B,$o,aa(A,fun(B,$o),P2,aa(nat,A,nth(A,Xs),P)),aa(nat,B,nth(B,Ys),P)) ) ) ).

% list_all2_nthD2
tff(fact_7958_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),P: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(B,$o,aa(A,fun(B,$o),P2,aa(nat,A,nth(A,Xs),P)),aa(nat,B,nth(B,Ys),P)) ) ) ).

% list_all2_nthD
tff(fact_7959_list__all2__append2,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),Zs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),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) )
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Us2),Ys)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Vs2),Zs) ) ) ).

% list_all2_append2
tff(fact_7960_list__all2__append1,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(A),Zs: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),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) )
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Us2)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Ys),Vs2) ) ) ).

% list_all2_append1
tff(fact_7961_list__all2__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P2: fun(A,fun(B,$o)),Us: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),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))
      <=> ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Us),Vs) ) ) ) ).

% list_all2_append
tff(fact_7962_list__all2__same,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xs: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,P2),Xs),Xs)
    <=> ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P2,X4),X4) ) ) ).

% list_all2_same
tff(fact_7963_list_Orel__refl__strong,axiom,
    ! [A: $tType,Xb: list(A),Ra2: fun(A,fun(A,$o))] :
      ( ! [Z2: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),Ra2,Z2),Z2) )
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,Ra2),Xb),Xb) ) ).

% list.rel_refl_strong
tff(fact_7964_list_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Xb: list(A),Yb: list(B),Ra2: fun(A,fun(B,$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),Xb),Yb)
     => ( ! [Z2: A,Yb2: B] :
            ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
           => ( member(B,Yb2,aa(list(B),set(B),set2(B),Yb))
             => ( aa(B,$o,aa(A,fun(B,$o),R,Z2),Yb2)
               => aa(B,$o,aa(A,fun(B,$o),Ra2,Z2),Yb2) ) ) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Ra2),Xb),Yb) ) ) ).

% list.rel_mono_strong
tff(fact_7965_list_Orel__cong,axiom,
    ! [A: $tType,B: $tType,Xb: list(A),Yaa: list(A),Yb: list(B),Xaa: list(B),R: fun(A,fun(B,$o)),Ra2: fun(A,fun(B,$o))] :
      ( ( Xb = Yaa )
     => ( ( Yb = Xaa )
       => ( ! [Z2: A,Yb2: B] :
              ( member(A,Z2,aa(list(A),set(A),set2(A),Yaa))
             => ( member(B,Yb2,aa(list(B),set(B),set2(B),Xaa))
               => ( aa(B,$o,aa(A,fun(B,$o),R,Z2),Yb2)
                <=> aa(B,$o,aa(A,fun(B,$o),Ra2,Z2),Yb2) ) ) )
         => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),Xb),Yb)
          <=> aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Ra2),Yaa),Xaa) ) ) ) ) ).

% list.rel_cong
tff(fact_7966_list__all2__lengthD,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),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_7967_list__all2__dropI,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),Nb: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
     => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),drop(A,Nb,Xs)),drop(B,Nb,Ys)) ) ).

% list_all2_dropI
tff(fact_7968_list__all2__takeI,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),Nb: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
     => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),take(A,Nb,Xs)),take(B,Nb,Ys)) ) ).

% list_all2_takeI
tff(fact_7969_list_Octr__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o))] : aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),nil(A)),nil(B)) ).

% list.ctr_transfer(1)
tff(fact_7970_list_Orel__distinct_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Y21: A,Y22: list(A)] : ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y22)),nil(B)) ).

% list.rel_distinct(2)
tff(fact_7971_list_Orel__distinct_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Y21: B,Y22: list(B)] : ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y22)) ).

% list.rel_distinct(1)
tff(fact_7972_list_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A2: list(A),B2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),A2),B2)
     => ( ( ( A2 = nil(A) )
         => ( B2 != nil(B) ) )
       => ~ ! [X15: A,X22: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),X22) )
             => ! [Y15: B,Y23: list(B)] :
                  ( ( B2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y15),Y23) )
                 => ( aa(B,$o,aa(A,fun(B,$o),R,X15),Y15)
                   => ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),X22),Y23) ) ) ) ) ) ).

% list.rel_cases
tff(fact_7973_list_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Xb: list(A),Yb: list(B),Q: fun(list(A),fun(list(B),$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),Xb),Yb)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Q,nil(A)),nil(B))
       => ( ! [A21: A,A222: list(A),B21: B,B222: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),R,A21),B21)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),Q,A222),B222)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),Q,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A21),A222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B21),B222)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),Q,Xb),Yb) ) ) ) ).

% list.rel_induct
tff(fact_7974_list__all2__induct,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),R: fun(list(A),fun(list(B),$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),R,nil(A)),nil(B))
       => ( ! [X3: A,Xs2: list(A),Y: B,Ys3: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),P2,X3),Y)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs2),Ys3)
               => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),R,Xs2),Ys3)
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3)) ) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),R,Xs),Ys) ) ) ) ).

% list_all2_induct
tff(fact_7975_list__all2__Cons2,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Yb: B,Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Yb),Ys))
    <=> ? [Z5: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z5),Zs3) )
          & aa(B,$o,aa(A,fun(B,$o),P2,Z5),Yb)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Zs3),Ys) ) ) ).

% list_all2_Cons2
tff(fact_7976_list__all2__Cons1,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xb: A,Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Ys)
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & aa(B,$o,aa(A,fun(B,$o),P2,Xb),Z5)
          & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Zs3) ) ) ).

% list_all2_Cons1
tff(fact_7977_list__all2__Cons,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xb: A,Xs: list(A),Yb: B,Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Yb),Ys))
    <=> ( aa(B,$o,aa(A,fun(B,$o),P2,Xb),Yb)
        & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys) ) ) ).

% list_all2_Cons
tff(fact_7978_list_Orel__intros_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),X21: A,Y21: B,X222: list(A),Y22: list(B)] :
      ( aa(B,$o,aa(A,fun(B,$o),R,X21),Y21)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),X222),Y22)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),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),Y22)) ) ) ).

% list.rel_intros(2)
tff(fact_7979_list_Orel__inject_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),X21: A,X222: list(A),Y21: B,Y22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),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),Y22))
    <=> ( aa(B,$o,aa(A,fun(B,$o),R,X21),Y21)
        & aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),X222),Y22) ) ) ).

% list.rel_inject(2)
tff(fact_7980_list_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o)),Ra2: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R),Ra2)
     => aa(fun(list(A),fun(list(B),$o)),$o,aa(fun(list(A),fun(list(B),$o)),fun(fun(list(A),fun(list(B),$o)),$o),ord_less_eq(fun(list(A),fun(list(B),$o))),list_all2(A,B,R)),list_all2(A,B,Ra2)) ) ).

% list.rel_mono
tff(fact_7981_list__all2__update__cong,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),Xb: A,Yb: B,I2: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
     => ( aa(B,$o,aa(A,fun(B,$o),P2,Xb),Yb)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),list_update(A,Xs,I2,Xb)),list_update(B,Ys,I2,Yb)) ) ) ).

% list_all2_update_cong
tff(fact_7982_list__all2__antisym,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Q: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( ! [X3: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),P2,X3),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),Q,Y),X3)
           => ( X3 = Y ) ) )
     => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,P2),Xs),Ys)
       => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,Q),Ys),Xs)
         => ( Xs = Ys ) ) ) ) ).

% list_all2_antisym
tff(fact_7983_list__all2__trans,axiom,
    ! [B: $tType,A: $tType,C: $tType,P1: fun(A,fun(B,$o)),P22: fun(B,fun(C,$o)),P32: fun(A,fun(C,$o)),As: list(A),Bs: list(B),Cs: list(C)] :
      ( ! [A4: A,B4: B,C2: C] :
          ( aa(B,$o,aa(A,fun(B,$o),P1,A4),B4)
         => ( aa(C,$o,aa(B,fun(C,$o),P22,B4),C2)
           => aa(C,$o,aa(A,fun(C,$o),P32,A4),C2) ) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P1),As),Bs)
       => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),list_all2(B,C,P22),Bs),Cs)
         => aa(list(C),$o,aa(list(A),fun(list(C),$o),list_all2(A,C,P32),As),Cs) ) ) ) ).

% list_all2_trans
tff(fact_7984_list__all2__refl,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xs: list(A)] :
      ( ! [X3: A] : aa(A,$o,aa(A,fun(A,$o),P2,X3),X3)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,P2),Xs),Xs) ) ).

% list_all2_refl
tff(fact_7985_list__all2__mono,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B),Q: fun(A,fun(B,$o))] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
     => ( ! [Xs2: A,Ys3: B] :
            ( aa(B,$o,aa(A,fun(B,$o),P2,Xs2),Ys3)
           => aa(B,$o,aa(A,fun(B,$o),Q,Xs2),Ys3) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,Q),Xs),Ys) ) ) ).

% list_all2_mono
tff(fact_7986_list__all2__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,fequal(A)),Xs),Ys) ) ).

% list_all2_eq
tff(fact_7987_list_Orel__refl,axiom,
    ! [A: $tType,Ra2: fun(A,fun(A,$o)),Xb: list(A)] :
      ( ! [X3: A] : aa(A,$o,aa(A,fun(A,$o),Ra2,X3),X3)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),list_all2(A,A,Ra2),Xb),Xb) ) ).

% list.rel_refl
tff(fact_7988_list_Orel__eq,axiom,
    ! [A: $tType] : list_all2(A,A,fequal(A)) = fequal(list(A)) ).

% list.rel_eq
tff(fact_7989_list__all2__appendI,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),A2: list(A),B2: list(B),C3: list(A),D3: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),A2),B2)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),C3),D3)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),C3)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),B2),D3)) ) ) ).

% list_all2_appendI
tff(fact_7990_list_Orel__sel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A2: list(A),B2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),A2),B2)
    <=> ( ( ( A2 = nil(A) )
        <=> ( B2 = nil(B) ) )
        & ( ( A2 != nil(A) )
         => ( ( B2 != nil(B) )
           => ( aa(B,$o,aa(A,fun(B,$o),R,aa(list(A),A,hd(A),A2)),aa(list(B),B,hd(B),B2))
              & aa(list(B),$o,aa(list(A),fun(list(B),$o),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_7991_subset__chain__insert,axiom,
    ! [A: $tType,A11: set(set(A)),B5: set(A),B11: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A11,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),B5),B11))
    <=> ( member(set(A),B5,A11)
        & ! [X4: set(A)] :
            ( member(set(A),X4,B11)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),B5)
              | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),X4) ) )
        & aa(set(set(A)),$o,pred_chain(set(A),A11,ord_less(set(A))),B11) ) ) ).

% subset_chain_insert
tff(fact_7992_subset__chain__def,axiom,
    ! [A: $tType,A11: set(set(A)),C10: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A11,ord_less(set(A))),C10)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C10),A11)
        & ! [X4: set(A)] :
            ( member(set(A),X4,C10)
           => ! [Xa2: set(A)] :
                ( member(set(A),Xa2,C10)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),Xa2)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa2),X4) ) ) ) ) ) ).

% subset_chain_def
tff(fact_7993_subset__Zorn,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ! [C6: set(set(A))] :
          ( aa(set(set(A)),$o,pred_chain(set(A),A5,ord_less(set(A))),C6)
         => ? [X: set(A)] :
              ( member(set(A),X,A5)
              & ! [Xa3: set(A)] :
                  ( member(set(A),Xa3,C6)
                 => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa3),X) ) ) )
     => ? [X3: set(A)] :
          ( member(set(A),X3,A5)
          & ! [Xa: set(A)] :
              ( member(set(A),Xa,A5)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa)
               => ( Xa = X3 ) ) ) ) ) ).

% subset_Zorn
tff(fact_7994_subset__Zorn_H,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ! [C6: set(set(A))] :
          ( aa(set(set(A)),$o,pred_chain(set(A),A5,ord_less(set(A))),C6)
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6),A5) )
     => ? [X3: set(A)] :
          ( member(set(A),X3,A5)
          & ! [Xa: set(A)] :
              ( member(set(A),Xa,A5)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa)
               => ( Xa = X3 ) ) ) ) ) ).

% subset_Zorn'
tff(fact_7995_subset_Ochain__empty,axiom,
    ! [A: $tType,A5: set(set(A))] : aa(set(set(A)),$o,pred_chain(set(A),A5,ord_less(set(A))),bot_bot(set(set(A)))) ).

% subset.chain_empty
tff(fact_7996_pred__on_Ochain__empty,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,fun(A,$o))] : aa(set(A),$o,pred_chain(A,A5,P2),bot_bot(set(A))) ).

% pred_on.chain_empty
tff(fact_7997_subset_OchainI,axiom,
    ! [A: $tType,C5: set(set(A)),A5: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C5),A5)
     => ( ! [X3: set(A),Y: set(A)] :
            ( member(set(A),X3,C5)
           => ( member(set(A),Y,C5)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X3),Y)
                | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Y),X3) ) ) )
       => aa(set(set(A)),$o,pred_chain(set(A),A5,ord_less(set(A))),C5) ) ) ).

% subset.chainI
tff(fact_7998_subset_Ochain__def,axiom,
    ! [A: $tType,A5: set(set(A)),C5: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A5,ord_less(set(A))),C5)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C5),A5)
        & ! [X4: set(A)] :
            ( member(set(A),X4,C5)
           => ! [Xa2: set(A)] :
                ( member(set(A),Xa2,C5)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X4),Xa2)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Xa2),X4) ) ) ) ) ) ).

% subset.chain_def
tff(fact_7999_subset_Ochain__total,axiom,
    ! [A: $tType,A5: set(set(A)),C5: set(set(A)),Xb: set(A),Yb: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A5,ord_less(set(A))),C5)
     => ( member(set(A),Xb,C5)
       => ( member(set(A),Yb,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Xb),Yb)
            | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Yb),Xb) ) ) ) ) ).

% subset.chain_total
tff(fact_8000_product__lists__set,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_akx(list(list(A)),fun(list(A),$o),Xss)) ).

% product_lists_set
tff(fact_8001_list__all2I,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),P2: fun(A,fun(B,$o))] :
      ( ! [X3: product_prod(A,B)] :
          ( 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,A2,B2)))
         => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P2),X3) )
     => ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),A2),B2) ) ) ).

% list_all2I
tff(fact_8002_subset__Zorn__nonempty,axiom,
    ! [A: $tType,A11: set(set(A))] :
      ( ( A11 != bot_bot(set(set(A))) )
     => ( ! [C11: set(set(A))] :
            ( ( C11 != bot_bot(set(set(A))) )
           => ( aa(set(set(A)),$o,pred_chain(set(A),A11,ord_less(set(A))),C11)
             => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C11),A11) ) )
       => ? [X3: set(A)] :
            ( member(set(A),X3,A11)
            & ! [Xa: set(A)] :
                ( member(set(A),Xa,A11)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa)
                 => ( Xa = X3 ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_8003_Union__in__chain,axiom,
    ! [A: $tType,B11: set(set(A)),A11: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),B11)
     => ( ( B11 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A11,ord_less(set(A))),B11)
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11),B11) ) ) ) ).

% Union_in_chain
tff(fact_8004_Inter__in__chain,axiom,
    ! [A: $tType,B11: set(set(A)),A11: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),B11)
     => ( ( B11 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A11,ord_less(set(A))),B11)
         => member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11),B11) ) ) ) ).

% Inter_in_chain
tff(fact_8005_subset_Ochain__extend,axiom,
    ! [A: $tType,A5: set(set(A)),C5: set(set(A)),Z: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A5,ord_less(set(A))),C5)
     => ( member(set(A),Z,A5)
       => ( ! [X3: set(A)] :
              ( member(set(A),X3,C5)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X3),Z) )
         => aa(set(set(A)),$o,pred_chain(set(A),A5,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))),insert(set(A)),Z),bot_bot(set(set(A))))),C5)) ) ) ) ).

% subset.chain_extend
tff(fact_8006_pred__on_Ochain__extend,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,fun(A,$o)),C5: set(A),Z: A] :
      ( aa(set(A),$o,pred_chain(A,A5,P2),C5)
     => ( member(A,Z,A5)
       => ( ! [X3: A] :
              ( member(A,X3,C5)
             => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P2),fequal(A)),X3),Z) )
         => aa(set(A),$o,pred_chain(A,A5,P2),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)),insert(A),Z),bot_bot(set(A)))),C5)) ) ) ) ).

% pred_on.chain_extend
tff(fact_8007_finite__subset__Union__chain,axiom,
    ! [A: $tType,A5: set(A),B11: set(set(A)),A11: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11))
       => ( ( B11 != bot_bot(set(set(A))) )
         => ( aa(set(set(A)),$o,pred_chain(set(A),A11,ord_less(set(A))),B11)
           => ~ ! [B8: set(A)] :
                  ( member(set(A),B8,B11)
                 => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B8) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_8008_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P2),Xs),Ys)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X4: product_prod(A,B)] :
            ( member(product_prod(A,B),X4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
           => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P2),X4) ) ) ) ).

% list_all2_iff
tff(fact_8009_Chains__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),chains(A,R2)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_yb(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))) ).

% Chains_subset
tff(fact_8010_Chains__alt__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R2)
     => ( chains(A,R2) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_yb(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))) ) ) ).

% Chains_alt_def
tff(fact_8011_chain__subset__alt__def,axiom,
    ! [A: $tType,C5: set(set(A))] :
      ( chain_subset(A,C5)
    <=> aa(set(set(A)),$o,pred_chain(set(A),top_top(set(set(A))),ord_less(set(A))),C5) ) ).

% chain_subset_alt_def
tff(fact_8012_power_Opower__eq__if,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),P: A,Mb: nat] :
      power2(A,One,Times,P,Mb) = $ite(Mb = zero_zero(nat),One,aa(A,A,aa(A,fun(A,A),Times,P),power2(A,One,Times,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))) ).

% power.power_eq_if
tff(fact_8013_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_8014_cauchyD,axiom,
    ! [X5: fun(nat,rat),R2: rat] :
      ( cauchy(X5)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
       => ? [K: nat] :
          ! [M3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M3)
           => ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N4)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X5,M3)),aa(nat,rat,X5,N4)))),R2) ) ) ) ) ).

% cauchyD
tff(fact_8015_cauchyI,axiom,
    ! [X5: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R3)
         => ? [K4: nat] :
            ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),M)
             => ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N2)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X5,M)),aa(nat,rat,X5,N2)))),R3) ) ) )
     => cauchy(X5) ) ).

% cauchyI
tff(fact_8016_cauchy__imp__bounded,axiom,
    ! [X5: fun(nat,rat)] :
      ( cauchy(X5)
     => ? [B4: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B4)
          & ! [N4: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N4))),B4) ) ) ).

% cauchy_imp_bounded
tff(fact_8017_cauchy__def,axiom,
    ! [X5: fun(nat,rat)] :
      ( cauchy(X5)
    <=> ! [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
         => ? [K3: nat] :
            ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),M2)
             => ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(nat,rat,X5,M2)),aa(nat,rat,X5,N)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_8018_le__Real,axiom,
    ! [X5: fun(nat,rat),Y5: fun(nat,rat)] :
      ( cauchy(X5)
     => ( cauchy(Y5)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real2(X5)),real2(Y5))
        <=> ! [R5: rat] :
              ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
             => ? [K3: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X5,N)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y5,N)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_8019_cauchy__not__vanishes,axiom,
    ! [X5: fun(nat,rat)] :
      ( cauchy(X5)
     => ( ~ vanishes(X5)
       => ? [B4: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B4)
            & ? [K: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N4)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B4),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N4))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_8020_vanishes__mult__bounded,axiom,
    ! [X5: fun(nat,rat),Y5: fun(nat,rat)] :
      ( ? [A9: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A9)
          & ! [N2: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N2))),A9) )
     => ( vanishes(Y5)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aky(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X5),Y5)) ) ) ).

% vanishes_mult_bounded
tff(fact_8021_vanishes__def,axiom,
    ! [X5: fun(nat,rat)] :
      ( vanishes(X5)
    <=> ! [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
         => ? [K3: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N))),R5) ) ) ) ).

% vanishes_def
tff(fact_8022_vanishesI,axiom,
    ! [X5: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R3)
         => ? [K4: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N2)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N2))),R3) ) )
     => vanishes(X5) ) ).

% vanishesI
tff(fact_8023_vanishesD,axiom,
    ! [X5: fun(nat,rat),R2: rat] :
      ( vanishes(X5)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
       => ? [K: nat] :
          ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N4)
           => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N4))),R2) ) ) ) ).

% vanishesD
tff(fact_8024_cauchy__not__vanishes__cases,axiom,
    ! [X5: fun(nat,rat)] :
      ( cauchy(X5)
     => ( ~ vanishes(X5)
       => ? [B4: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B4)
            & ? [K: nat] :
                ( ! [N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N4)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B4),aa(rat,rat,uminus_uminus(rat),aa(nat,rat,X5,N4))) )
                | ! [N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N4)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B4),aa(nat,rat,X5,N4)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_8025_not__positive__Real,axiom,
    ! [X5: fun(nat,rat)] :
      ( cauchy(X5)
     => ( ~ aa(real,$o,positive2,real2(X5))
      <=> ! [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
           => ? [K3: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X5,N)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_8026_positive__Real,axiom,
    ! [X5: fun(nat,rat)] :
      ( cauchy(X5)
     => ( aa(real,$o,positive2,real2(X5))
      <=> ? [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
            & ? [K3: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,X5,N)) ) ) ) ) ).

% positive_Real
tff(fact_8027_less__real__def,axiom,
    ! [Xb: real,Yb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Yb)
    <=> aa(real,$o,positive2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Yb),Xb)) ) ).

% less_real_def
tff(fact_8028_Real_Opositive_Orep__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,positive2,Xb)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K3: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,aa(real,fun(nat,rat),rep_real,Xb),N)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_8029_finite__def,axiom,
    ! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),$o),aTP_Lamp_zz(fun(set(A),$o),fun(set(A),$o))) ).

% finite_def
tff(fact_8030_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F3)
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F3)) = complete_lattice_lfp(A,F3) ) ) ) ).

% lfp_funpow
tff(fact_8031_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),K2: nat] :
          ( order_mono(A,A,F3)
         => ( ( 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,K2)),F3),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)),K2),F3),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K2),F3),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_8032_lfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P2: fun(A,$o)] :
          ( order_mono(A,A,F3)
         => ( ! [S6: A] :
                ( aa(A,$o,P2,S6)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S6),complete_lattice_lfp(A,F3))
                 => aa(A,$o,P2,aa(A,A,F3,S6)) ) )
           => ( ! [M7: set(A)] :
                  ( ! [X: A] :
                      ( member(A,X,M7)
                     => aa(A,$o,P2,X) )
                 => aa(A,$o,P2,aa(set(A),A,complete_Sup_Sup(A),M7)) )
             => aa(A,$o,P2,complete_lattice_lfp(A,F3)) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_8033_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: A,F3: fun(A,A),P2: A] :
          ( ( A5 = complete_lattice_lfp(A,F3) )
         => ( order_mono(A,A,F3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),P2))),P2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),P2) ) ) ) ) ).

% def_lfp_induct
tff(fact_8034_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: fun($o,A),Y5: fun($o,A)] :
          ( aa(fun($o,A),$o,aa(fun($o,A),fun(fun($o,A),$o),ord_less_eq(fun($o,A)),X5),Y5)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,X5,$false)),aa($o,A,Y5,$false))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,X5,$true)),aa($o,A,Y5,$true)) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_8035_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,F3: fun(set(product_prod(A,B)),set(product_prod(A,B))),P2: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),complete_lattice_lfp(set(product_prod(A,B)),F3))
     => ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F3)
       => ( ! [A4: A,B4: B] :
              ( 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)),F3,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)),F3)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P2)))))
             => aa(B,$o,aa(A,fun(B,$o),P2,A4),B4) )
         => aa(B,$o,aa(A,fun(B,$o),P2,A2),B2) ) ) ) ).

% lfp_induct2
tff(fact_8036_lfp__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),A5: A] :
          ( ! [U3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F3,U3)),U3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),U3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),complete_lattice_lfp(A,F3)) ) ) ).

% lfp_greatest
tff(fact_8037_lfp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),A5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F3,A5)),A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F3)),A5) ) ) ).

% lfp_lowerbound
tff(fact_8038_lfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),G3: fun(A,A)] :
          ( ! [Z7: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F3,Z7)),aa(A,A,G3,Z7))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F3)),complete_lattice_lfp(A,G3)) ) ) ).

% lfp_mono
tff(fact_8039_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,fun(A,A))] :
          ( ! [X3: A,Y: A,W: A,Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),W)),aa(A,A,aa(A,fun(A,A),F3,Y),Z2)) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_akz(fun(A,fun(A,A)),fun(A,A),F3)) = complete_lattice_lfp(A,aTP_Lamp_ala(fun(A,fun(A,A)),fun(A,A),F3)) ) ) ) ).

% lfp_lfp
tff(fact_8040_lfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F4: fun(A,A),Xb: A] :
          ( order_mono(A,A,F4)
         => ( ( aa(A,A,F4,Xb) = Xb )
           => ( ! [Z2: A] :
                  ( ( aa(A,A,F4,Z2) = Z2 )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z2) )
             => ( complete_lattice_lfp(A,F4) = Xb ) ) ) ) ) ).

% lfp_eqI
tff(fact_8041_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] : complete_lattice_lfp(A,F3) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_alb(fun(A,A),fun(A,$o),F3))) ) ).

% lfp_def
tff(fact_8042_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),P2: A] :
          ( order_mono(A,A,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F3)),P2))),P2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F3)),P2) ) ) ) ).

% lfp_induct
tff(fact_8043_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: fun(A,A)] : comple7512665784863727008ratesp(A,X) = complete_lattice_lfp(fun(A,$o),aTP_Lamp_aaw(fun(A,A),fun(fun(A,$o),fun(A,$o)),X)) ) ).

% iteratesp_def
tff(fact_8044_Real_Opositive__def,axiom,
    positive2 = aa(fun(fun(nat,rat),$o),fun(real,$o),map_fun(real,fun(nat,rat),$o,$o,rep_real,id($o)),aTP_Lamp_alc(fun(nat,rat),$o)) ).

% Real.positive_def
tff(fact_8045_iteratesp_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M4: set(A),F3: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M4)
         => ( ! [X3: A] :
                ( member(A,X3,M4)
               => aa(A,$o,comple7512665784863727008ratesp(A,F3),X3) )
           => aa(A,$o,comple7512665784863727008ratesp(A,F3),aa(set(A),A,complete_Sup_Sup(A),M4)) ) ) ) ).

% iteratesp.Sup
tff(fact_8046_iteratesp_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),A2: A] :
          ( aa(A,$o,comple7512665784863727008ratesp(A,F3),A2)
         => ( ! [X3: A] :
                ( ( A2 = aa(A,A,F3,X3) )
               => ~ aa(A,$o,comple7512665784863727008ratesp(A,F3),X3) )
           => ~ ! [M7: set(A)] :
                  ( ( A2 = aa(set(A),A,complete_Sup_Sup(A),M7) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M7)
                   => ~ ! [X: A] :
                          ( member(A,X,M7)
                         => aa(A,$o,comple7512665784863727008ratesp(A,F3),X) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_8047_iteratesp_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A),A2: A] :
          ( aa(A,$o,comple7512665784863727008ratesp(A,F3),A2)
        <=> ( ? [X4: A] :
                ( ( A2 = aa(A,A,F3,X4) )
                & aa(A,$o,comple7512665784863727008ratesp(A,F3),X4) )
            | ? [M8: set(A)] :
                ( ( A2 = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X4: A] :
                    ( member(A,X4,M8)
                   => aa(A,$o,comple7512665784863727008ratesp(A,F3),X4) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_8048_lfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P2: fun(A,$o),F3: fun(A,A),Alpha: fun(A,B),G3: fun(B,B)] :
          ( aa(A,$o,P2,bot_bot(A))
         => ( ! [X3: A] :
                ( aa(A,$o,P2,X3)
               => aa(A,$o,P2,aa(A,A,F3,X3)) )
           => ( ! [M7: fun(nat,A)] :
                  ( ! [I: nat] : aa(A,$o,P2,aa(nat,A,M7,I))
                 => aa(A,$o,P2,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,M7),top_top(set(nat))))) )
             => ( ! [M7: fun(nat,A)] :
                    ( order_mono(nat,A,M7)
                   => ( ! [I: nat] : aa(A,$o,P2,aa(nat,A,M7,I))
                     => ( aa(A,B,Alpha,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,M7),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_ald(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M7)),top_top(set(nat)))) ) ) )
               => ( order_sup_continuous(A,A,F3)
                 => ( order_sup_continuous(B,B,G3)
                   => ( ! [X3: A] :
                          ( aa(A,$o,P2,X3)
                         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),complete_lattice_lfp(A,F3))
                           => ( aa(A,B,Alpha,aa(A,A,F3,X3)) = aa(B,B,G3,aa(A,B,Alpha,X3)) ) ) )
                     => ( ! [X3: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G3,X3))
                       => ( aa(A,B,Alpha,complete_lattice_lfp(A,F3)) = complete_lattice_lfp(B,G3) ) ) ) ) ) ) ) ) ) ) ).

% lfp_transfer_bounded
tff(fact_8049_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_ale(fun(A,A),fun(nat,A),F4)),top_top(set(nat)))) ) ) ) ).

% sup_continuous_lfp
tff(fact_8050_lfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Alpha: fun(A,B),F3: fun(A,A),G3: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( order_sup_continuous(A,A,F3)
           => ( order_sup_continuous(B,B,G3)
             => ( ! [X3: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G3,X3))
               => ( ! [X3: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),complete_lattice_lfp(A,F3))
                     => ( aa(A,B,Alpha,aa(A,A,F3,X3)) = aa(B,B,G3,aa(A,B,Alpha,X3)) ) )
                 => ( aa(A,B,Alpha,complete_lattice_lfp(A,F3)) = complete_lattice_lfp(B,G3) ) ) ) ) ) ) ) ).

% lfp_transfer
tff(fact_8051_cclfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta3822494911875563373attice(A) )
     => ! [Alpha: fun(A,B),F3: fun(A,A),G3: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( order_mono(A,A,F3)
           => ( ( aa(A,B,Alpha,bot_bot(A)) = bot_bot(B) )
             => ( ! [X3: A] : aa(A,B,Alpha,aa(A,A,F3,X3)) = aa(B,B,G3,aa(A,B,Alpha,X3))
               => ( aa(A,B,Alpha,order_532582986084564980_cclfp(A,F3)) = order_532582986084564980_cclfp(B,G3) ) ) ) ) ) ) ).

% cclfp_transfer
tff(fact_8052_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_aej(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ) ).

% ord_class.lexordp_def
tff(fact_8053_finite__refines__card__le,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A5,R))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S2)
       => ( equiv_equiv(A,A5,R)
         => ( equiv_equiv(A,A5,S2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A5,S2))),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A5,R))) ) ) ) ) ).

% finite_refines_card_le
tff(fact_8054_lexordp__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),nil(A)),Ys)
        <=> ( Ys != nil(A) ) ) ) ).

% lexordp_simps(1)
tff(fact_8055_lexordp__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] : ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),nil(A)) ) ).

% lexordp_simps(2)
tff(fact_8056_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A),Yb: A,Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
              & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_8057_quotient__disj,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),X5: set(A),Y5: set(A)] :
      ( equiv_equiv(A,A5,R2)
     => ( member(set(A),X5,equiv_quotient(A,A5,R2))
       => ( member(set(A),Y5,equiv_quotient(A,A5,R2))
         => ( ( X5 = Y5 )
            | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X5),Y5) = bot_bot(set(A)) ) ) ) ) ) ).

% quotient_disj
tff(fact_8058_equiv__listrel,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( equiv_equiv(A,A5,R2)
     => equiv_equiv(list(A),lists(A,A5),listrel(A,A,R2)) ) ).

% equiv_listrel
tff(fact_8059_lexordp__append__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ys: list(A),Xs: list(A)] :
          ( ( Ys != nil(A) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),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_8060_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A),Us: list(A),Vs: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Vs))
         => ( ! [A4: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),A4)
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Us),Vs) ) ) ) ).

% lexordp_append_leftD
tff(fact_8061_lexordp__append__leftI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Us: list(A),Vs: list(A),Xs: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Us),Vs)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),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_8062_lexordp__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
         => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Ys),Xs) ) ) ).

% lexordp_antisym
tff(fact_8063_lexordp__trans,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A),Zs: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
         => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Ys),Zs)
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Zs) ) ) ) ).

% lexordp_trans
tff(fact_8064_lexordp__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
          | ( Xs = Ys )
          | aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Ys),Xs) ) ) ).

% lexordp_linear
tff(fact_8065_lexordp__irreflexive_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xs: list(A)] : ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Xs) ) ).

% lexordp_irreflexive'
tff(fact_8066_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),X3)
         => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Xs) ) ) ).

% lexordp_irreflexive
tff(fact_8067_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Yb: A,Xs: list(A),Ys: list(A)] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Yb),Xb)
           => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_8068_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Yb: A,Xs: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) ) ) ).

% lexordp.Cons
tff(fact_8069_lexordp_ONil,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Yb: A,Ys: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) ) ).

% lexordp.Nil
tff(fact_8070_in__quotient__imp__non__empty,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( equiv_equiv(A,A5,R2)
     => ( member(set(A),X5,equiv_quotient(A,A5,R2))
       => ( X5 != bot_bot(set(A)) ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_8071_in__quotient__imp__closed,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),X5: set(A),Xb: A,Yb: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(set(A),X5,equiv_quotient(A,A5,R2))
       => ( member(A,Xb,X5)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
           => member(A,Yb,X5) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_8072_quotient__eq__iff,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),X5: set(A),Y5: set(A),Xb: A,Yb: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(set(A),X5,equiv_quotient(A,A5,R2))
       => ( member(set(A),Y5,equiv_quotient(A,A5,R2))
         => ( member(A,Xb,X5)
           => ( member(A,Yb,Y5)
             => ( ( X5 = Y5 )
              <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_8073_quotient__eqI,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),X5: set(A),Y5: set(A),Xb: A,Yb: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(set(A),X5,equiv_quotient(A,A5,R2))
       => ( member(set(A),Y5,equiv_quotient(A,A5,R2))
         => ( member(A,Xb,X5)
           => ( member(A,Yb,Y5)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_8074_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
         => ( ( ( A1 = nil(A) )
             => ! [Y: A,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3) )
           => ( ! [X3: A] :
                  ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)
                 => ! [Y: A] :
                      ( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y) ) )
             => ~ ! [X3: A,Y: A,Xs2: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                   => ! [Ys3: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3) )
                       => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X3)
                           => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_8075_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X4)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs3),Ys4) ) ) ) ) ).

% lexordp.simps
tff(fact_8076_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
         => ( ( ( Xs = nil(A) )
             => ! [Y: A,Ys5: list(A)] : Ys != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys5) )
           => ( ! [X3: A] :
                  ( ? [Xs4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4)
                 => ! [Y: A] :
                      ( ? [Ys5: list(A)] : Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys5)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y) ) )
             => ~ ! [X3: A,Xs4: list(A)] :
                    ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
                   => ! [Ys5: list(A)] :
                        ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys5) )
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs4),Ys5) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_8077_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A),P2: fun(list(A),fun(list(A),$o))] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
         => ( ! [Y: A,Ys3: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3))
           => ( ! [X3: A,Xs2: list(A),Y: A,Ys3: list(A)] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                 => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)) )
             => ( ! [X3: A,Xs2: list(A),Ys3: list(A)] :
                    ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3)
                   => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,Xs2),Ys3)
                     => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3)) ) )
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P2,Xs),Ys) ) ) ) ) ) ).

% lexordp_induct
tff(fact_8078_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Yb: A,Us: list(A),Xs: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Yb)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),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),Xb),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),Yb),Ys))) ) ) ).

% lexordp_append_left_rightI
tff(fact_8079_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
        <=> ( ? [X4: A,Vs2: list(A)] : Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Vs2))
            | ? [Us2: list(A),A7: A,B7: A,Vs2: list(A),Ws3: list(A)] :
                ( aa(A,$o,aa(A,fun(A,$o),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_8080_finite__refines__finite,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A5,R))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S2)
       => ( equiv_equiv(A,A5,R)
         => ( equiv_equiv(A,A5,S2)
           => aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A5,S2)) ) ) ) ) ).

% finite_refines_finite
tff(fact_8081_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,Yb: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(A,Xb,A5)
       => ( member(A,Yb,A5)
         => ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))),R2) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),bot_bot(set(A))),R2) )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_8082_equiv__imp__dvd__card,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),K2: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( equiv_equiv(A,A5,R2)
       => ( ! [X7: set(A)] :
              ( member(set(A),X7,equiv_quotient(A,A5,R2))
             => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),aa(set(A),nat,finite_card(A),X7)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K2),aa(set(A),nat,finite_card(A),A5)) ) ) ) ).

% equiv_imp_dvd_card
tff(fact_8083_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),X5: set(A),Xb: A,Yb: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(set(A),X5,equiv_quotient(A,A5,R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),bot_bot(set(A))))),X5)
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_8084_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
        <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),ord_less(A))))) ) ) ).

% lexordp_conv_lexord
tff(fact_8085_UN__equiv__class__inject,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R2: set(product_prod(A,A)),F3: fun(A,set(B)),X5: set(A),Y5: set(A)] :
      ( equiv_equiv(A,A5,R2)
     => ( equiv_congruent(A,set(B),R2,F3)
       => ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),X5)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),Y5)) )
         => ( member(set(A),X5,equiv_quotient(A,A5,R2))
           => ( member(set(A),Y5,equiv_quotient(A,A5,R2))
             => ( ! [X3: A,Y: A] :
                    ( member(A,X3,A5)
                   => ( member(A,Y,A5)
                     => ( ( aa(A,set(B),F3,X3) = aa(A,set(B),F3,Y) )
                       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),R2) ) ) )
               => ( X5 = Y5 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_8086_proj__iff,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,Yb: A] :
      ( equiv_equiv(A,A5,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),bot_bot(set(A))))),A5)
       => ( ( equiv_proj(A,A,R2,Xb) = equiv_proj(A,A,R2,Yb) )
        <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2) ) ) ) ).

% proj_iff
tff(fact_8087_congruentI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F3: fun(A,B)] :
      ( ! [Y: A,Z2: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
         => ( aa(A,B,F3,Y) = aa(A,B,F3,Z2) ) )
     => equiv_congruent(A,B,R2,F3) ) ).

% congruentI
tff(fact_8088_congruentD,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),F3: fun(A,B),Yb: A,Z: A] :
      ( equiv_congruent(A,B,R2,F3)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Yb),Z),R2)
       => ( aa(A,B,F3,Yb) = aa(A,B,F3,Z) ) ) ) ).

% congruentD
tff(fact_8089_ord_Olexordp__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : lexordp2(A,Less) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_aei(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Less)) ).

% ord.lexordp_def
tff(fact_8090_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R2: set(product_prod(A,A)),F3: fun(A,set(B)),A2: A] :
      ( equiv_equiv(A,A5,R2)
     => ( equiv_congruent(A,set(B),R2,F3)
       => ( member(A,A2,A5)
         => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F3),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) = aa(A,set(B),F3,A2) ) ) ) ) ).

% UN_equiv_class
tff(fact_8091_ImageI,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,R2: set(product_prod(A,B)),A5: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),R2)
     => ( member(A,A2,A5)
       => member(B,B2,image(A,B,R2,A5)) ) ) ).

% ImageI
tff(fact_8092_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_8093_ord_Olexordp__simps_I3_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xb: A,Xs: list(A),Yb: A,Ys: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys))
    <=> ( aa(A,$o,aa(A,fun(A,$o),Less,Xb),Yb)
        | ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Yb),Xb)
          & aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs),Ys) ) ) ) ).

% ord.lexordp_simps(3)
tff(fact_8094_ord_Olexordp__simps_I1_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Ys: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),nil(A)),Ys)
    <=> ( Ys != nil(A) ) ) ).

% ord.lexordp_simps(1)
tff(fact_8095_ord_Olexordp__simps_I2_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xs: list(A)] : ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs),nil(A)) ).

% ord.lexordp_simps(2)
tff(fact_8096_Image__empty1,axiom,
    ! [B: $tType,A: $tType,X5: set(B)] : image(B,A,bot_bot(set(product_prod(B,A))),X5) = bot_bot(set(A)) ).

% Image_empty1
tff(fact_8097_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A2: B] :
      ( member(A,B2,image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B)))))
    <=> 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_8098_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))),insert(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))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listrel_Nil
tff(fact_8099_equiv__class__self,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),A2: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(A,A2,A5)
       => member(A,A2,image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ) ).

% equiv_class_self
tff(fact_8100_proj__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),Xb: B] : equiv_proj(B,A,R2,Xb) = image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xb),bot_bot(set(B)))) ).

% proj_def
tff(fact_8101_countable__Image,axiom,
    ! [B: $tType,A: $tType,Y5: set(A),X5: set(product_prod(A,B))] :
      ( ! [Y: A] :
          ( member(A,Y,Y5)
         => countable_countable(B,image(A,B,X5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))) )
     => ( countable_countable(A,Y5)
       => countable_countable(B,image(A,B,X5,Y5)) ) ) ).

% countable_Image
tff(fact_8102_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A2: B] : image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B)))) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_alf(set(product_prod(B,A)),fun(B,fun(A,$o)),R2),A2)) ).

% Image_singleton
tff(fact_8103_finite__rtrancl__Image,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(set(A),$o,finite_finite2(A),A5)
       => aa(set(A),$o,finite_finite2(A),image(A,A,transitive_rtrancl(A,R),A5)) ) ) ).

% finite_rtrancl_Image
tff(fact_8104_ord_Olexordp__append__rightI,axiom,
    ! [A: $tType,Ys: list(A),Less: fun(A,fun(A,$o)),Xs: list(A)] :
      ( ( Ys != nil(A) )
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),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_8105_ord_Olexordp__append__left__rightI,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xb: A,Yb: A,Us: list(A),Xs: list(A),Ys: list(A)] :
      ( aa(A,$o,aa(A,fun(A,$o),Less,Xb),Yb)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),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),Xb),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),Yb),Ys))) ) ).

% ord.lexordp_append_left_rightI
tff(fact_8106_ord_Olexordp__append__leftI,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Us: list(A),Vs: list(A),Xs: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Us),Vs)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),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_8107_ord_Olexordp__append__leftD,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xs: list(A),Us: list(A),Vs: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),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] : ~ aa(A,$o,aa(A,fun(A,$o),Less,A4),A4)
       => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Us),Vs) ) ) ).

% ord.lexordp_append_leftD
tff(fact_8108_ord_Olexordp_Ocong,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : lexordp2(A,Less) = lexordp2(A,Less) ).

% ord.lexordp.cong
tff(fact_8109_ord_Olexordp__irreflexive,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xs: list(A)] :
      ( ! [X3: A] : ~ aa(A,$o,aa(A,fun(A,$o),Less,X3),X3)
     => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs),Xs) ) ).

% ord.lexordp_irreflexive
tff(fact_8110_finite__Image,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),A5: set(A)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => aa(set(B),$o,finite_finite2(B),image(A,B,R,A5)) ) ).

% finite_Image
tff(fact_8111_ord_Olexordp_OCons,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xb: A,Yb: A,Xs: list(A),Ys: list(A)] :
      ( aa(A,$o,aa(A,fun(A,$o),Less,Xb),Yb)
     => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) ) ).

% ord.lexordp.Cons
tff(fact_8112_ord_Olexordp_OCons__eq,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xb: A,Yb: A,Xs: list(A),Ys: list(A)] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Xb),Yb)
     => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Yb),Xb)
       => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs),Ys)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) ) ) ) ).

% ord.lexordp.Cons_eq
tff(fact_8113_ord_Olexordp_ONil,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Yb: A,Ys: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Yb),Ys)) ).

% ord.lexordp.Nil
tff(fact_8114_ord_Olexordp_Ocases,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),A1: list(A),A22: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),A1),A22)
     => ( ( ( A1 = nil(A) )
         => ! [Y: A,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3) )
       => ( ! [X3: A] :
              ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)
             => ! [Y: A] :
                  ( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)
                 => ~ aa(A,$o,aa(A,fun(A,$o),Less,X3),Y) ) )
         => ~ ! [X3: A,Y: A,Xs2: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ! [Ys3: list(A)] :
                    ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3) )
                   => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,X3),Y)
                     => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y),X3)
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs2),Ys3) ) ) ) ) ) ) ) ).

% ord.lexordp.cases
tff(fact_8115_ord_Olexordp_Osimps,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),A1: list(A),A22: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),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) ) )
        | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & aa(A,$o,aa(A,fun(A,$o),Less,X4),Y4) )
        | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & ~ aa(A,$o,aa(A,fun(A,$o),Less,X4),Y4)
            & ~ aa(A,$o,aa(A,fun(A,$o),Less,Y4),X4)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs3),Ys4) ) ) ) ).

% ord.lexordp.simps
tff(fact_8116_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A2: A,A5: set(A),B2: B,R2: set(product_prod(A,B))] :
      ( member(A,A2,A5)
     => ( 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)
       => member(B,B2,image(A,B,R2,A5)) ) ) ).

% rev_ImageI
tff(fact_8117_Image__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A5: set(B)] :
      ( member(A,B2,image(B,A,R2,A5))
    <=> ? [X4: B] :
          ( member(B,X4,A5)
          & 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) ) ) ).

% Image_iff
tff(fact_8118_ImageE,axiom,
    ! [A: $tType,B: $tType,B2: A,R2: set(product_prod(B,A)),A5: set(B)] :
      ( member(A,B2,image(B,A,R2,A5))
     => ~ ! [X3: B] :
            ( 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)
           => ~ member(B,X3,A5) ) ) ).

% ImageE
tff(fact_8119_Image__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S: set(B)] : image(B,A,R2,S) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_alg(set(product_prod(B,A)),fun(set(B),fun(A,$o)),R2),S)) ).

% Image_def
tff(fact_8120_quotientI,axiom,
    ! [A: $tType,Xb: A,A5: set(A),R2: set(product_prod(A,A))] :
      ( member(A,Xb,A5)
     => member(set(A),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))),equiv_quotient(A,A5,R2)) ) ).

% quotientI
tff(fact_8121_quotientE,axiom,
    ! [A: $tType,X5: set(A),A5: set(A),R2: set(product_prod(A,A))] :
      ( member(set(A),X5,equiv_quotient(A,A5,R2))
     => ~ ! [X3: A] :
            ( ( X5 = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A)))) )
           => ~ member(A,X3,A5) ) ) ).

% quotientE
tff(fact_8122_wfI__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),image(A,A,R,A6))
         => ( A6 = bot_bot(set(A)) ) )
     => wf(A,R) ) ).

% wfI_pf
tff(fact_8123_wfE__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( wf(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),image(A,A,R,A5))
       => ( A5 = bot_bot(set(A)) ) ) ) ).

% wfE_pf
tff(fact_8124_listrel__Cons,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),Xb: 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))),insert(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),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)),insert(B),Xb),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))),insert(list(B)),Xs),bot_bot(set(list(B)))))) ).

% listrel_Cons
tff(fact_8125_equiv__class__eq__iff,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,Yb: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2)
      <=> ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),bot_bot(set(A)))) )
          & member(A,Xb,A5)
          & member(A,Yb,A5) ) ) ) ).

% equiv_class_eq_iff
tff(fact_8126_eq__equiv__class__iff,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,Yb: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(A,Xb,A5)
       => ( member(A,Yb,A5)
         => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Yb),bot_bot(set(A)))) )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Yb),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_8127_equiv__class__eq,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( equiv_equiv(A,A5,R2)
     => ( 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)),insert(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_8128_eq__equiv__class,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A,A5: set(A)] :
      ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A5,R2)
       => ( member(A,B2,A5)
         => 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_8129_refines__equiv__class__eq2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A)),A5: set(A),A2: A] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S2)
     => ( equiv_equiv(A,A5,R)
       => ( equiv_equiv(A,A5,S2)
         => ( image(A,A,S2,image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_8130_refines__equiv__class__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A)),A5: set(A),A2: A] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S2)
     => ( equiv_equiv(A,A5,R)
       => ( equiv_equiv(A,A5,S2)
         => ( image(A,A,R,image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_8131_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)
     => ( member(A,A2,field2(A,R2))
       => ( member(A,B2,field2(A,R2))
         => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_8132_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),B5: set(B)] : image(B,A,R2,B5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_alh(set(product_prod(B,A)),fun(B,set(A)),R2)),B5)) ).

% Image_eq_UN
tff(fact_8133_equiv__class__subset,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) ) ) ).

% equiv_class_subset
tff(fact_8134_subset__equiv__class,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),B2: A,A2: A] :
      ( equiv_equiv(A,A5,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))
       => ( member(A,B2,A5)
         => 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_8135_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,A2: A,B2: A] :
      ( equiv_equiv(A,A5,R2)
     => ( member(A,Xb,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)),insert(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_8136_singleton__quotient,axiom,
    ! [A: $tType,Xb: A,R2: set(product_prod(A,A))] : equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))),R2) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xb),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% singleton_quotient
tff(fact_8137_quotient__def,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] : equiv_quotient(A,A5,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_ali(set(product_prod(A,A)),fun(A,set(set(A))),R2)),A5)) ).

% quotient_def
tff(fact_8138_Image__fold,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(A)] :
      ( aa(set(product_prod(A,B)),$o,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_alj(set(A),fun(A,fun(B,fun(set(B),set(B)))),S2)),bot_bot(set(B)),R) ) ) ).

% Image_fold
tff(fact_8139_congruent2__implies__congruent__UN,axiom,
    ! [A: $tType,C: $tType,B: $tType,A14: set(A),R1: set(product_prod(A,A)),A25: set(B),R22: set(product_prod(B,B)),F3: fun(A,fun(B,set(C))),A2: B] :
      ( equiv_equiv(A,A14,R1)
     => ( equiv_equiv(B,A25,R22)
       => ( equiv_congruent2(A,B,set(C),R1,R22,F3)
         => ( 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_alk(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F3),A2)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_8140_UN__equiv__class2,axiom,
    ! [A: $tType,C: $tType,B: $tType,A14: set(A),R1: set(product_prod(A,A)),A25: set(B),R22: set(product_prod(B,B)),F3: fun(A,fun(B,set(C))),A1: A,A22: B] :
      ( equiv_equiv(A,A14,R1)
     => ( equiv_equiv(B,A25,R22)
       => ( equiv_congruent2(A,B,set(C),R1,R22,F3)
         => ( member(A,A1,A14)
           => ( 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_alk(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F3),A22)),image(A,A,R1,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F3,A1),A22) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_8141_congruent2I_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( ! [Y15: A,Z12: A,Y23: B,Z23: B] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y15),Z12),R1)
         => ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y23),Z23),R22)
           => ( aa(B,C,aa(A,fun(B,C),F3,Y15),Y23) = aa(B,C,aa(A,fun(B,C),F3,Z12),Z23) ) ) )
     => equiv_congruent2(A,B,C,R1,R22,F3) ) ).

% congruent2I'
tff(fact_8142_congruent2D,axiom,
    ! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F3: fun(A,fun(B,C)),Y1: A,Z1: A,Y2: B,Z22: B] :
      ( equiv_congruent2(A,B,C,R1,R22,F3)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z1),R1)
       => ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z22),R22)
         => ( aa(B,C,aa(A,fun(B,C),F3,Y1),Y2) = aa(B,C,aa(A,fun(B,C),F3,Z1),Z22) ) ) ) ) ).

% congruent2D
tff(fact_8143_congruent2__commuteI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R2: set(product_prod(A,A)),F3: fun(A,fun(A,B))] :
      ( equiv_equiv(A,A5,R2)
     => ( ! [Y: A,Z2: A] :
            ( member(A,Y,A5)
           => ( member(A,Z2,A5)
             => ( aa(A,B,aa(A,fun(A,B),F3,Y),Z2) = aa(A,B,aa(A,fun(A,B),F3,Z2),Y) ) ) )
       => ( ! [Y: A,Z2: A,W: A] :
              ( member(A,W,A5)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R2)
               => ( aa(A,B,aa(A,fun(A,B),F3,W),Y) = aa(A,B,aa(A,fun(A,B),F3,W),Z2) ) ) )
         => equiv_congruent2(A,A,B,R2,R2,F3) ) ) ) ).

% congruent2_commuteI
tff(fact_8144_congruent2I,axiom,
    ! [C: $tType,B: $tType,A: $tType,A14: set(A),R1: set(product_prod(A,A)),A25: set(B),R22: set(product_prod(B,B)),F3: fun(A,fun(B,C))] :
      ( equiv_equiv(A,A14,R1)
     => ( equiv_equiv(B,A25,R22)
       => ( ! [Y: A,Z2: A,W: B] :
              ( member(B,W,A25)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R1)
               => ( aa(B,C,aa(A,fun(B,C),F3,Y),W) = aa(B,C,aa(A,fun(B,C),F3,Z2),W) ) ) )
         => ( ! [Y: B,Z2: B,W: A] :
                ( member(A,W,A14)
               => ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y),Z2),R22)
                 => ( aa(B,C,aa(A,fun(B,C),F3,W),Y) = aa(B,C,aa(A,fun(B,C),F3,W),Z2) ) ) )
           => equiv_congruent2(A,B,C,R1,R22,F3) ) ) ) ) ).

% congruent2I
tff(fact_8145_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( order_preorder_on(A,field2(A,R2),R2)
     => ( member(A,A2,field2(A,R2))
       => ( member(A,B2,field2(A,R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_8146_subset__Image__Image__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A),B5: set(A)] :
      ( order_preorder_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),field2(A,R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),field2(A,R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,A5)),image(A,A,R2,B5))
          <=> ! [X4: A] :
                ( member(A,X4,A5)
               => ? [Xa2: A] :
                    ( member(A,Xa2,B5)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R2) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_8147_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_8148_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)
       => ( member(A,A2,field2(A,R2))
         => ( member(A,B2,field2(A,R2))
           => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) )
            <=> ( A2 = B2 ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_8149_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat,S: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(A,set(A),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),aa(A,set(A),set_ord_lessThan(A),S))),Nb) = aa(nat,A,infini527867602293511546merate(A,S2),Nb) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_8150_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Mb: nat,Nb: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),Mb)),aa(nat,A,infini527867602293511546merate(A,S2),Nb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% enumerate_mono_iff
tff(fact_8151_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Mb: nat,Nb: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(set(A),nat,finite_card(A),S2))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S2))
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),Mb)),aa(nat,A,infini527867602293511546merate(A,S2),Nb))
              <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_8152_le__enumerate,axiom,
    ! [S2: set(nat),Nb: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,infini527867602293511546merate(nat,S2),Nb)) ) ).

% le_enumerate
tff(fact_8153_antisym__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisym(A,R2)
    <=> ! [X4: A,Y4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X4),R2)
           => ( X4 = Y4 ) ) ) ) ).

% antisym_def
tff(fact_8154_antisymI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X3: A,Y: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X3),R2)
           => ( X3 = Y ) ) )
     => antisym(A,R2) ) ).

% antisymI
tff(fact_8155_antisymD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( antisym(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2),R2)
         => ( A2 = B2 ) ) ) ) ).

% antisymD
tff(fact_8156_antisym__empty,axiom,
    ! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).

% antisym_empty
tff(fact_8157_enumerate__Ex,axiom,
    ! [S2: set(nat),S: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => ( member(nat,S,S2)
       => ? [N2: nat] : aa(nat,nat,infini527867602293511546merate(nat,S2),N2) = S ) ) ).

% enumerate_Ex
tff(fact_8158_enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => member(A,aa(nat,A,infini527867602293511546merate(A,S2),Nb),S2) ) ) ).

% enumerate_in_set
tff(fact_8159_strict__mono__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => order_strict_mono(nat,nat,infini527867602293511546merate(nat,S2)) ) ).

% strict_mono_enumerate
tff(fact_8160_antisym__singleton,axiom,
    ! [A: $tType,Xb: 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))),insert(product_prod(A,A)),Xb),bot_bot(set(product_prod(A,A))))) ).

% antisym_singleton
tff(fact_8161_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),Nb)),aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Nb))) ) ) ).

% enumerate_step
tff(fact_8162_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Mb: nat,Nb: nat,S2: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( ~ aa(set(A),$o,finite_finite2(A),S2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),Mb)),aa(nat,A,infini527867602293511546merate(A,S2),Nb)) ) ) ) ).

% enumerate_mono
tff(fact_8163_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X5: set(A),Y5: set(A)] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X5))
             => ( aa(nat,A,infini527867602293511546merate(A,X5),I3) = aa(nat,A,infini527867602293511546merate(A,Y5),I3) ) )
         => ( aa(set(A),$o,finite_finite2(A),X5)
           => ( aa(set(A),$o,finite_finite2(A),Y5)
             => ( ( aa(set(A),nat,finite_card(A),X5) = aa(set(A),nat,finite_card(A),Y5) )
               => ( X5 = Y5 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_8164_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),S: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( member(A,S,S2)
           => ? [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),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_8165_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S2))
           => member(A,aa(nat,A,infini527867602293511546merate(A,S2),Nb),S2) ) ) ) ).

% finite_enumerate_in_set
tff(fact_8166_range__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => ( aa(set(nat),set(nat),image2(nat,nat,infini527867602293511546merate(nat,S2)),top_top(set(nat))) = S2 ) ) ).

% range_enumerate
tff(fact_8167_inj__enumerate,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => inj_on(nat,A,infini527867602293511546merate(A,S2),top_top(set(nat))) ) ) ).

% inj_enumerate
tff(fact_8168_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Mb: nat,Nb: nat,S2: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(A),nat,finite_card(A),S2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),Mb)),aa(nat,A,infini527867602293511546merate(A,S2),Nb)) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_8169_finite__le__enumerate,axiom,
    ! [S2: set(nat),Nb: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(set(nat),nat,finite_card(nat),S2))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,infini527867602293511546merate(nat,S2),Nb)) ) ) ).

% finite_le_enumerate
tff(fact_8170_bij__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => bij_betw(nat,nat,infini527867602293511546merate(nat,S2),top_top(set(nat)),S2) ) ).

% bij_enumerate
tff(fact_8171_finite__bij__enumerate,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => bij_betw(nat,A,infini527867602293511546merate(A,S2),aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),S2)),S2) ) ) ).

% finite_bij_enumerate
tff(fact_8172_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),S2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),Nb)),aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Nb))) ) ) ) ).

% finite_enumerate_step
tff(fact_8173_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] : aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Nb)) = 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)),insert(A),aa(nat,A,infini527867602293511546merate(A,S2),zero_zero(nat))),bot_bot(set(A))))),Nb) ) ).

% enumerate_Suc'
tff(fact_8174_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X5: set(A),Y5: set(A)] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(set(A),nat,finite_card(A),X5))
             => ( aa(nat,A,infini527867602293511546merate(A,X5),I3) = aa(nat,A,infini527867602293511546merate(A,Y5),I3) ) )
         => ( aa(set(A),$o,finite_finite2(A),X5)
           => ( aa(set(A),$o,finite_finite2(A),Y5)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X5)),aa(set(A),nat,finite_card(A),Y5))
               => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),Y5) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_8175_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),S2))
           => ( aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Nb)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_all(set(A),fun(nat,fun(A,$o)),S2),Nb)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_8176_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] : aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Nb)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),ord_Least(A,aTP_Lamp_alm(set(A),fun(A,$o),S2))),bot_bot(set(A))))),Nb) ) ).

% enumerate_Suc
tff(fact_8177_Least__eq__0,axiom,
    ! [P2: fun(nat,$o)] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ord_Least(nat,P2) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_8178_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] : aa(nat,A,infini527867602293511546merate(A,S2),zero_zero(nat)) = ord_Least(A,aTP_Lamp_alm(set(A),fun(A,$o),S2)) ) ).

% enumerate_0
tff(fact_8179_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
         => ( ? [X_1: A] : aa(A,$o,P2,X_1)
           => ( ord_Least(A,P2) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),P2)) ) ) ) ) ).

% Least_Min
tff(fact_8180_Least__Suc,axiom,
    ! [P2: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P2,Nb)
     => ( ~ aa(nat,$o,P2,zero_zero(nat))
       => ( ord_Least(nat,P2) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_aln(fun(nat,$o),fun(nat,$o),P2))) ) ) ) ).

% Least_Suc
tff(fact_8181_Least__Suc2,axiom,
    ! [P2: fun(nat,$o),Nb: nat,Q: fun(nat,$o),Mb: nat] :
      ( aa(nat,$o,P2,Nb)
     => ( aa(nat,$o,Q,Mb)
       => ( ~ aa(nat,$o,P2,zero_zero(nat))
         => ( ! [K: nat] :
                ( aa(nat,$o,P2,aa(nat,nat,suc,K))
              <=> aa(nat,$o,Q,K) )
           => ( ord_Least(nat,P2) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_8182_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o),A2: A,Q: fun(A,$o)] :
          ( aa(A,$o,P2,A2)
         => ( ! [X3: A] :
                ( aa(A,$o,P2,X3)
               => aa(A,$o,Q,X3) )
           => aa(A,$o,Q,ord_Least(A,P2)) ) ) ) ).

% LeastI2
tff(fact_8183_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P2,X_1)
         => aa(A,$o,P2,ord_Least(A,P2)) ) ) ).

% LeastI_ex
tff(fact_8184_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P2,X_1)
         => ( ! [X3: A] :
                ( aa(A,$o,P2,X3)
               => aa(A,$o,Q,X3) )
           => aa(A,$o,Q,ord_Least(A,P2)) ) ) ) ).

% LeastI2_ex
tff(fact_8185_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_dr(nat,real),Uu) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_1
tff(fact_8186_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rs(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_8187_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_kl(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).

% ATP.lambda_3
tff(fact_8188_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_ns(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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_4
tff(fact_8189_ATP_Olambda__5,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_zu(A,$o),Uu)
        <=> ( member(A,Uu,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Uu) ) ) ) ).

% ATP.lambda_5
tff(fact_8190_ATP_Olambda__6,axiom,
    ! [A: $tType,Uu: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_akf(set(A),$o),Uu)
    <=> ( ( Uu != bot_bot(set(A)) )
        & countable_countable(A,Uu) ) ) ).

% ATP.lambda_6
tff(fact_8191_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [Uu: product_prod(A,A)] : aa(product_prod(A,A),A,aTP_Lamp_xv(product_prod(A,A),A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(product_prod(A,A),A,product_fst(A,A),Uu)),aa(product_prod(A,A),A,product_snd(A,A),Uu)) ) ).

% ATP.lambda_7
tff(fact_8192_ATP_Olambda__8,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_adg(list(A),product_prod(nat,list(A))),Uu) = aa(list(A),product_prod(nat,list(A)),aa(nat,fun(list(A),product_prod(nat,list(A))),product_Pair(nat,list(A)),aa(list(A),nat,size_size(list(A)),Uu)),Uu) ).

% ATP.lambda_8
tff(fact_8193_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( aa(nat,$o,aTP_Lamp_ll(nat,$o),Uu)
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_9
tff(fact_8194_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_yf(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_10
tff(fact_8195_ATP_Olambda__11,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_yj(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_11
tff(fact_8196_ATP_Olambda__12,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_yk(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_12
tff(fact_8197_ATP_Olambda__13,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_db(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_13
tff(fact_8198_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_aek(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_14
tff(fact_8199_ATP_Olambda__15,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_mu(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_15
tff(fact_8200_ATP_Olambda__16,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_acz(product_prod(int,int),$o),Uu)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_16
tff(fact_8201_ATP_Olambda__17,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_sj(nat,A),Uu) = divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_17
tff(fact_8202_ATP_Olambda__18,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ahe(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_18
tff(fact_8203_ATP_Olambda__19,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_abq(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))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_19
tff(fact_8204_ATP_Olambda__20,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_ahf(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_20
tff(fact_8205_ATP_Olambda__21,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_agn(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_21
tff(fact_8206_ATP_Olambda__22,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_adv(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_adu(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_22
tff(fact_8207_ATP_Olambda__23,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_adl(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_adk(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_23
tff(fact_8208_ATP_Olambda__24,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_yq(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_yp(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_24
tff(fact_8209_ATP_Olambda__25,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_ox(real,real),Uu) = suminf(real,aTP_Lamp_hr(real,fun(nat,real),Uu)) ).

% ATP.lambda_25
tff(fact_8210_ATP_Olambda__26,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_qf(nat,set(nat)),Uu) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ap(nat,fun(nat,$o),Uu)) ).

% ATP.lambda_26
tff(fact_8211_ATP_Olambda__27,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_ze(fun(A,B),set(product_prod(A,B))),Uu) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_zd(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_27
tff(fact_8212_ATP_Olambda__28,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(real,real)),aTP_Lamp_xm(real,filter(product_prod(real,real))),Uu) = principal(product_prod(real,real),aa(fun(product_prod(real,real),$o),set(product_prod(real,real)),collect(product_prod(real,real)),aa(fun(real,fun(real,$o)),fun(product_prod(real,real),$o),product_case_prod(real,real,$o),aTP_Lamp_xl(real,fun(real,fun(real,$o)),Uu)))) ).

% ATP.lambda_28
tff(fact_8213_ATP_Olambda__29,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(complex,complex)),aTP_Lamp_xk(real,filter(product_prod(complex,complex))),Uu) = principal(product_prod(complex,complex),aa(fun(product_prod(complex,complex),$o),set(product_prod(complex,complex)),collect(product_prod(complex,complex)),aa(fun(complex,fun(complex,$o)),fun(product_prod(complex,complex),$o),product_case_prod(complex,complex,$o),aTP_Lamp_xj(real,fun(complex,fun(complex,$o)),Uu)))) ).

% ATP.lambda_29
tff(fact_8214_ATP_Olambda__30,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real] : aa(real,filter(product_prod(A,A)),aTP_Lamp_xh(real,filter(product_prod(A,A))),Uu) = principal(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_xg(real,fun(A,fun(A,$o)),Uu)))) ) ).

% ATP.lambda_30
tff(fact_8215_ATP_Olambda__31,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_sf(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_31
tff(fact_8216_ATP_Olambda__32,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ahd(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_32
tff(fact_8217_ATP_Olambda__33,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_agw(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_33
tff(fact_8218_ATP_Olambda__34,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bi(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_34
tff(fact_8219_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_aav(A,$o),Uu)
        <=> ? [N: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N) ) ) ) ).

% ATP.lambda_35
tff(fact_8220_ATP_Olambda__36,axiom,
    ! [Uu: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aTP_Lamp_alc(fun(nat,rat),$o),Uu)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K3: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,Uu,N)) ) ) ) ).

% ATP.lambda_36
tff(fact_8221_ATP_Olambda__37,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zy(product_prod(A,A),$o),Uu)
        <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ) ).

% ATP.lambda_37
tff(fact_8222_ATP_Olambda__38,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_akn(product_prod(A,A),$o),Uu)
    <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).

% ATP.lambda_38
tff(fact_8223_ATP_Olambda__39,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_aay(real,$o),Uu)
    <=> ? [I4: int,N: nat] :
          ( ( Uu = divide_divide(real,aa(int,real,ring_1_of_int(real),I4),aa(nat,real,semiring_1_of_nat(real),N)) )
          & ( N != zero_zero(nat) ) ) ) ).

% ATP.lambda_39
tff(fact_8224_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_aaf(product_prod(A,A),$o),Uu)
        <=> ? [X4: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4) ) ) ) ).

% ATP.lambda_40
tff(fact_8225_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_aag(product_prod(A,A),$o),Uu)
        <=> ? [X4: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X4) ) ) ) ).

% ATP.lambda_41
tff(fact_8226_ATP_Olambda__42,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_aai(product_prod(A,A),$o),Uu)
        <=> ? [X4: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4) ) ) ) ).

% ATP.lambda_42
tff(fact_8227_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_aah(product_prod(A,A),$o),Uu)
        <=> ? [X4: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X4) ) ) ) ).

% ATP.lambda_43
tff(fact_8228_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_aae(product_prod(A,A),$o),Uu)
        <=> ? [X4: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4) )
              & ( X4 != Y4 ) ) ) ) ).

% ATP.lambda_44
tff(fact_8229_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_iq(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_45
tff(fact_8230_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_ff(nat,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_46
tff(fact_8231_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_ip(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_47
tff(fact_8232_ATP_Olambda__48,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_adz(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = $ite(aa(product_prod(A,C),C,product_snd(A,C),Uu) = aa(product_prod(C,B),C,product_fst(C,B),Uua),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_48
tff(fact_8233_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_fe(nat,fun(nat,A),Uu),Uua) = $ite(~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_49
tff(fact_8234_ATP_Olambda__50,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aTP_Lamp_akc(set(A),fun(fun(A,nat),$o),Uu),Uua)
    <=> $ite(aa(set(A),$o,finite_finite2(A),Uu),bij_betw(A,nat,Uua,Uu,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),Uu))),bij_betw(A,nat,Uua,Uu,top_top(set(nat)))) ) ).

% ATP.lambda_50
tff(fact_8235_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_yd(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_51
tff(fact_8236_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_np(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_52
tff(fact_8237_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ale(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).

% ATP.lambda_53
tff(fact_8238_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ya(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).

% ATP.lambda_54
tff(fact_8239_ATP_Olambda__55,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_akx(list(list(A)),fun(list(A),$o),Uu),Uua)
    <=> aa(list(list(A)),$o,aa(list(A),fun(list(list(A)),$o),list_all2(A,list(A),aTP_Lamp_akw(A,fun(list(A),$o))),Uua),Uu) ) ).

% ATP.lambda_55
tff(fact_8240_ATP_Olambda__56,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_kh(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_56
tff(fact_8241_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ala(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_57
tff(fact_8242_ATP_Olambda__58,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_afi(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afh(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_58
tff(fact_8243_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fv(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_fu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_59
tff(fact_8244_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ft(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_fs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_60
tff(fact_8245_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_cy(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_61
tff(fact_8246_ATP_Olambda__62,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hv(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))))),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_62
tff(fact_8247_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_go(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_63
tff(fact_8248_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hb(nat,fun(nat,A),Uu),Uua) = divide_divide(A,aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_64
tff(fact_8249_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ir(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uua),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_65
tff(fact_8250_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_cx(nat,fun(nat,a)),Uu),Uua) = $let(
        m3: a,
        m3:= aa(a,a,aa(a,fun(a,a),times_times(a),aa(num,a,numeral_numeral(a),bit0(one2))),aa(nat,a,semiring_1_of_nat(a),Uu)),
        $ite(Uua = zero_zero(nat),m3,aa(a,a,aa(a,fun(a,a),plus_plus(a),m3),one_one(a))) ) ).

% ATP.lambda_66
tff(fact_8251_ATP_Olambda__67,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ajp(complex,fun(real,$o),Uu),Uua)
    <=> ( ( sgn_sgn(complex,Uu) = cis(Uua) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi) ) ) ).

% ATP.lambda_67
tff(fact_8252_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_iu(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_68
tff(fact_8253_ATP_Olambda__69,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_iv(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_69
tff(fact_8254_ATP_Olambda__70,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hr(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))) ).

% ATP.lambda_70
tff(fact_8255_ATP_Olambda__71,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_oy(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_71
tff(fact_8256_ATP_Olambda__72,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ry(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_72
tff(fact_8257_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gr(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_73
tff(fact_8258_ATP_Olambda__74,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ali(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))),insert(set(A)),image(A,A,Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% ATP.lambda_74
tff(fact_8259_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gx(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_75
tff(fact_8260_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gv(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uu,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_76
tff(fact_8261_ATP_Olambda__77,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kn(nat,fun(nat,$o)),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_77
tff(fact_8262_ATP_Olambda__78,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_km(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_78
tff(fact_8263_ATP_Olambda__79,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_nt(set(option(A)),fun(option(A),$o),Uu),Uua)
    <=> ( member(option(A),Uua,Uu)
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_79
tff(fact_8264_ATP_Olambda__80,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aau(set(set(A)),fun(set(A),$o),Uu),Uua)
    <=> ( member(set(A),Uua,Uu)
        & ! [X4: set(A)] :
            ( member(set(A),X4,Uu)
           => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),X4) ) ) ) ).

% ATP.lambda_80
tff(fact_8265_ATP_Olambda__81,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hu(real,fun(nat,real),Uu),Uua) = divide_divide(real,aa(nat,real,power_power(real,Uu),Uua),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_81
tff(fact_8266_ATP_Olambda__82,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_wd(set(A),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_82
tff(fact_8267_ATP_Olambda__83,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_ot(set(A),fun(set(A),$o)),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uu),Uua)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_83
tff(fact_8268_ATP_Olambda__84,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yh(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,Uu),Uua)),Uua) ).

% ATP.lambda_84
tff(fact_8269_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fp(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).

% ATP.lambda_85
tff(fact_8270_ATP_Olambda__86,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fo(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uua) ).

% ATP.lambda_86
tff(fact_8271_ATP_Olambda__87,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_abm(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))),insert(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_87
tff(fact_8272_ATP_Olambda__88,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_ait(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))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_88
tff(fact_8273_ATP_Olambda__89,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_cn(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,power_power(complex,Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_89
tff(fact_8274_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ax(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,power_power(A,Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_90
tff(fact_8275_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_zt(A,fun(A,$o),Uu),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu) ) ) ) ).

% ATP.lambda_91
tff(fact_8276_ATP_Olambda__92,axiom,
    ! [A: $tType,Uu: fun(set(A),$o),Uua: set(A)] :
      ( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_zz(fun(set(A),$o),fun(set(A),$o)),Uu),Uua)
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A8: set(A),A7: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A7),A8) )
            & aa(set(A),$o,Uu,A8) ) ) ) ).

% ATP.lambda_92
tff(fact_8277_ATP_Olambda__93,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ajz(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)),insert(A),Uua),bot_bot(set(A))),Uu) ).

% ATP.lambda_93
tff(fact_8278_ATP_Olambda__94,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ie(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_94
tff(fact_8279_ATP_Olambda__95,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_xz(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ) ).

% ATP.lambda_95
tff(fact_8280_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_yi(nat,fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu)) ) ) ).

% ATP.lambda_96
tff(fact_8281_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_97
tff(fact_8282_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_98
tff(fact_8283_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_sk(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_99
tff(fact_8284_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_co(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_100
tff(fact_8285_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jd(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_101
tff(fact_8286_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jb(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_102
tff(fact_8287_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_eo(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_103
tff(fact_8288_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_104
tff(fact_8289_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_105
tff(fact_8290_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_106
tff(fact_8291_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cr(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_107
tff(fact_8292_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cq(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_108
tff(fact_8293_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_sl(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_109
tff(fact_8294_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dg(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_110
tff(fact_8295_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ve(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y4: A] :
                ( aa(A,$o,Uu,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Uua) ) ) ) ) ).

% ATP.lambda_111
tff(fact_8296_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fl(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)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_112
tff(fact_8297_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fi(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)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_113
tff(fact_8298_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_alb(fun(A,A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua) ) ) ).

% ATP.lambda_114
tff(fact_8299_ATP_Olambda__115,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_adh(fun(A,nat),fun(A,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(A,nat,Uu,Uua)),Uua) ).

% ATP.lambda_115
tff(fact_8300_ATP_Olambda__116,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_tr(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_116
tff(fact_8301_ATP_Olambda__117,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_aeb(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_117
tff(fact_8302_ATP_Olambda__118,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_lw(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_118
tff(fact_8303_ATP_Olambda__119,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_bz(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_119
tff(fact_8304_ATP_Olambda__120,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ee(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_120
tff(fact_8305_ATP_Olambda__121,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_sv(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ry(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_121
tff(fact_8306_ATP_Olambda__122,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_su(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ry(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_122
tff(fact_8307_ATP_Olambda__123,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_aee(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_aed(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_123
tff(fact_8308_ATP_Olambda__124,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_lk(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ).

% ATP.lambda_124
tff(fact_8309_ATP_Olambda__125,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_te(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_125
tff(fact_8310_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ik(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_126
tff(fact_8311_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_im(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,power_power(A,Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_127
tff(fact_8312_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_in(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_128
tff(fact_8313_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_em(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_129
tff(fact_8314_ATP_Olambda__130,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_nq(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Uua)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_130
tff(fact_8315_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_st(A,fun(nat,A),Uu),Uua) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_131
tff(fact_8316_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ss(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_132
tff(fact_8317_ATP_Olambda__133,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ib(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_133
tff(fact_8318_ATP_Olambda__134,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ij(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_134
tff(fact_8319_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_vj(A,fun(set(A),$o),Uu),Uua)
        <=> ( topolo1002775350975398744n_open(A,Uua)
            & member(A,Uu,Uua) ) ) ) ).

% ATP.lambda_135
tff(fact_8320_ATP_Olambda__136,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aco(set(A),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,finite_finite2(A),Uua)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ) ).

% ATP.lambda_136
tff(fact_8321_ATP_Olambda__137,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: list(product_prod(A,B))] :
      ( aa(list(product_prod(A,B)),$o,aTP_Lamp_akt(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),Uu),Uua)
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).

% ATP.lambda_137
tff(fact_8322_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lb(nat,fun(nat,A)),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_138
tff(fact_8323_ATP_Olambda__139,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_aec(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua)),product_lists(A,Uu)) ).

% ATP.lambda_139
tff(fact_8324_ATP_Olambda__140,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_adj(list(A),fun(list(A),$o)),Uu),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_140
tff(fact_8325_ATP_Olambda__141,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aka(set(A),fun(list(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu) ) ).

% ATP.lambda_141
tff(fact_8326_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ha(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_142
tff(fact_8327_ATP_Olambda__143,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_aef(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(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_143
tff(fact_8328_ATP_Olambda__144,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_agt(set(nat),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_8329_ATP_Olambda__145,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aen(set(nat),fun(nat,$o),Uu),Uua)
    <=> member(nat,aa(nat,nat,suc,Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_8330_ATP_Olambda__146,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_be(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),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_146
tff(fact_8331_ATP_Olambda__147,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_bd(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),bit0(one2))),Uua)) ) ).

% ATP.lambda_147
tff(fact_8332_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dq(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_148
tff(fact_8333_ATP_Olambda__149,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_abj(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)),insert(A),Uu)),Uua)) ).

% ATP.lambda_149
tff(fact_8334_ATP_Olambda__150,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_alh(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)),insert(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_150
tff(fact_8335_ATP_Olambda__151,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_aje(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)),insert(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_151
tff(fact_8336_ATP_Olambda__152,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_is(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_152
tff(fact_8337_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_so(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_153
tff(fact_8338_ATP_Olambda__154,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_afc(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_154
tff(fact_8339_ATP_Olambda__155,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_za(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_155
tff(fact_8340_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ke(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_156
tff(fact_8341_ATP_Olambda__157,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_acl(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_157
tff(fact_8342_ATP_Olambda__158,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_sp(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_158
tff(fact_8343_ATP_Olambda__159,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_yg(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ).

% ATP.lambda_159
tff(fact_8344_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_se(A,fun(nat,A),Uu),Uua) = divide_divide(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_160
tff(fact_8345_ATP_Olambda__161,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_afw(nat,fun(list(A),$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_161
tff(fact_8346_ATP_Olambda__162,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hq(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_162
tff(fact_8347_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jj(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_163
tff(fact_8348_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ei(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_164
tff(fact_8349_ATP_Olambda__165,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_akw(A,fun(list(A),$o)),Uu),Uua)
    <=> member(A,Uu,aa(list(A),set(A),set2(A),Uua)) ) ).

% ATP.lambda_165
tff(fact_8350_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_agk(list(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_166
tff(fact_8351_ATP_Olambda__167,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ahk(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_167
tff(fact_8352_ATP_Olambda__168,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_ae(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ).

% ATP.lambda_168
tff(fact_8353_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_169
tff(fact_8354_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_abd(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_170
tff(fact_8355_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ts(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_171
tff(fact_8356_ATP_Olambda__172,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_fg(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_172
tff(fact_8357_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_no(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_173
tff(fact_8358_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_lu(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_174
tff(fact_8359_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_yt(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_175
tff(fact_8360_ATP_Olambda__176,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_176
tff(fact_8361_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_abe(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_177
tff(fact_8362_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_tt(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_178
tff(fact_8363_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_dd(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_179
tff(fact_8364_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sd(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_180
tff(fact_8365_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mj(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_8366_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_lm(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_182
tff(fact_8367_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_lt(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_183
tff(fact_8368_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_oe(nat,fun(real,real),Uu),Uua) = aa(nat,real,power_power(real,Uua),Uu) ).

% ATP.lambda_184
tff(fact_8369_ATP_Olambda__185,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_mz(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_185
tff(fact_8370_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_afd(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_186
tff(fact_8371_ATP_Olambda__187,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_nc(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_187
tff(fact_8372_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ys(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_188
tff(fact_8373_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_yu(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_8374_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ls(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_190
tff(fact_8375_ATP_Olambda__191,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_og(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_191
tff(fact_8376_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ap(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),Uu) ) ).

% ATP.lambda_192
tff(fact_8377_ATP_Olambda__193,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_aj(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uua),Uu) ) ).

% ATP.lambda_193
tff(fact_8378_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ak(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uua),Uu) ) ) ).

% ATP.lambda_194
tff(fact_8379_ATP_Olambda__195,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_aic(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).

% ATP.lambda_195
tff(fact_8380_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_la(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_196
tff(fact_8381_ATP_Olambda__197,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_ym(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_197
tff(fact_8382_ATP_Olambda__198,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_afg(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_198
tff(fact_8383_ATP_Olambda__199,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_yl(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_199
tff(fact_8384_ATP_Olambda__200,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_aed(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_200
tff(fact_8385_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_zh(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_201
tff(fact_8386_ATP_Olambda__202,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_si(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_202
tff(fact_8387_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_alm(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_203
tff(fact_8388_ATP_Olambda__204,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_a(set(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_204
tff(fact_8389_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_xy(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_205
tff(fact_8390_ATP_Olambda__206,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_aer(list(A),fun(nat,list(A)),Uu),Uua) = drop(A,Uua,Uu) ).

% ATP.lambda_206
tff(fact_8391_ATP_Olambda__207,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_afv(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_207
tff(fact_8392_ATP_Olambda__208,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_cc(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_208
tff(fact_8393_ATP_Olambda__209,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aew(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_209
tff(fact_8394_ATP_Olambda__210,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_aex(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_210
tff(fact_8395_ATP_Olambda__211,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ut(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_211
tff(fact_8396_ATP_Olambda__212,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_uo(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_212
tff(fact_8397_ATP_Olambda__213,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ajk(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_213
tff(fact_8398_ATP_Olambda__214,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_aim(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),Uu) ).

% ATP.lambda_214
tff(fact_8399_ATP_Olambda__215,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_do(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))) ).

% ATP.lambda_215
tff(fact_8400_ATP_Olambda__216,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_dn(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)) ).

% ATP.lambda_216
tff(fact_8401_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_th(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,divide_divide(A,one_one(A),Uua)) ) ).

% ATP.lambda_217
tff(fact_8402_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_agr(fun(nat,$o),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_218
tff(fact_8403_ATP_Olambda__219,axiom,
    ! [A: $tType,Uu: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_ags(fun(nat,$o),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)) ) ).

% ATP.lambda_219
tff(fact_8404_ATP_Olambda__220,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: product_prod(B,A)] : aa(product_prod(B,A),nat,aTP_Lamp_zc(fun(A,nat),fun(product_prod(B,A),nat),Uu),Uua) = aa(A,nat,Uu,aa(product_prod(B,A),A,product_snd(B,A),Uua)) ).

% ATP.lambda_220
tff(fact_8405_ATP_Olambda__221,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,nat),Uua: product_prod(A,B)] : aa(product_prod(A,B),nat,aTP_Lamp_zb(fun(A,nat),fun(product_prod(A,B),nat),Uu),Uua) = aa(A,nat,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).

% ATP.lambda_221
tff(fact_8406_ATP_Olambda__222,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aln(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_222
tff(fact_8407_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_eu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_223
tff(fact_8408_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ev(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_224
tff(fact_8409_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ec(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_225
tff(fact_8410_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ci(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_226
tff(fact_8411_ATP_Olambda__227,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aff(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_227
tff(fact_8412_ATP_Olambda__228,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_add(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_228
tff(fact_8413_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_ack(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_229
tff(fact_8414_ATP_Olambda__230,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_acq(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_230
tff(fact_8415_ATP_Olambda__231,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_acr(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_231
tff(fact_8416_ATP_Olambda__232,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_afk(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_afj(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_232
tff(fact_8417_ATP_Olambda__233,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_lj(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_li(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_233
tff(fact_8418_ATP_Olambda__234,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_lh(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_lg(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_234
tff(fact_8419_ATP_Olambda__235,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_lf(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_le(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_235
tff(fact_8420_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_ld(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_lc(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_236
tff(fact_8421_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_kz(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_ky(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_237
tff(fact_8422_ATP_Olambda__238,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_oo(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_on(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_238
tff(fact_8423_ATP_Olambda__239,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_oh(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_239
tff(fact_8424_ATP_Olambda__240,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A] : aa(A,set(A),aTP_Lamp_nh(real,fun(A,set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ng(real,fun(A,fun(A,$o)),Uu),Uua)) ) ).

% ATP.lambda_240
tff(fact_8425_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_ajm(nat,fun(nat,complex),Uu),Uua) = cis(divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua)),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_241
tff(fact_8426_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_po(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_242
tff(fact_8427_ATP_Olambda__243,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ago(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_243
tff(fact_8428_ATP_Olambda__244,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_aiu(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_ait(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_244
tff(fact_8429_ATP_Olambda__245,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_aea(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_adz(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).

% ATP.lambda_245
tff(fact_8430_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_at(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Uu,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_246
tff(fact_8431_ATP_Olambda__247,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_nn(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_247
tff(fact_8432_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_hd(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_248
tff(fact_8433_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ds(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_249
tff(fact_8434_ATP_Olambda__250,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_agl(list(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_250
tff(fact_8435_ATP_Olambda__251,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aiz(nat,fun(nat,set(nat)),Uu),Uua) = aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ).

% ATP.lambda_251
tff(fact_8436_ATP_Olambda__252,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_sn(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_252
tff(fact_8437_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ob(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_253
tff(fact_8438_ATP_Olambda__254,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_yp(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_254
tff(fact_8439_ATP_Olambda__255,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_wh(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu,Uua)) ) ).

% ATP.lambda_255
tff(fact_8440_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_vx(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu)) ) ).

% ATP.lambda_256
tff(fact_8441_ATP_Olambda__257,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_abn(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))),insert(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_257
tff(fact_8442_ATP_Olambda__258,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_aiv(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))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_258
tff(fact_8443_ATP_Olambda__259,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ade(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_259
tff(fact_8444_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_adf(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_260
tff(fact_8445_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kb(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_261
tff(fact_8446_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ka(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_262
tff(fact_8447_ATP_Olambda__263,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_akh(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_263
tff(fact_8448_ATP_Olambda__264,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_agc(A,fun(A,$o),Uu),Uua)
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_264
tff(fact_8449_ATP_Olambda__265,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_agj(A,fun(A,$o),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_265
tff(fact_8450_ATP_Olambda__266,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_akz(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_266
tff(fact_8451_ATP_Olambda__267,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_hj(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_267
tff(fact_8452_ATP_Olambda__268,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_ra(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_268
tff(fact_8453_ATP_Olambda__269,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: A] : aa(A,B,aTP_Lamp_kq(fun(A,$o),fun(A,B),Uu),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uua)) ) ).

% ATP.lambda_269
tff(fact_8454_ATP_Olambda__270,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,set(B),aTP_Lamp_aio(fun(A,B),fun(A,set(B)),Uu),Uua) = aa(B,set(B),set_ord_atMost(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_270
tff(fact_8455_ATP_Olambda__271,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_sh(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_271
tff(fact_8456_ATP_Olambda__272,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_pm(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_272
tff(fact_8457_ATP_Olambda__273,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wo(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_273
tff(fact_8458_ATP_Olambda__274,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wu(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_274
tff(fact_8459_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_oa(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8460_ATP_Olambda__276,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vo(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8461_ATP_Olambda__277,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_277
tff(fact_8462_ATP_Olambda__278,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_qo(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8463_ATP_Olambda__279,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_dt(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_279
tff(fact_8464_ATP_Olambda__280,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_cu(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_280
tff(fact_8465_ATP_Olambda__281,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_cv(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_281
tff(fact_8466_ATP_Olambda__282,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_282
tff(fact_8467_ATP_Olambda__283,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_fd(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_283
tff(fact_8468_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ea(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_284
tff(fact_8469_ATP_Olambda__285,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_du(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_8470_ATP_Olambda__286,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_bq(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_286
tff(fact_8471_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_el(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_287
tff(fact_8472_ATP_Olambda__288,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rl(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_288
tff(fact_8473_ATP_Olambda__289,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_oz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_8474_ATP_Olambda__290,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vs(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8475_ATP_Olambda__291,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_rg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_291
tff(fact_8476_ATP_Olambda__292,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,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_8477_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wv(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_293
tff(fact_8478_ATP_Olambda__294,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vp(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_294
tff(fact_8479_ATP_Olambda__295,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_qz(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8480_ATP_Olambda__296,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_hx(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_296
tff(fact_8481_ATP_Olambda__297,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_hw(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_8482_ATP_Olambda__298,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wy(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_298
tff(fact_8483_ATP_Olambda__299,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8484_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_ol(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_8485_ATP_Olambda__301,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_qn(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8486_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_re(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_302
tff(fact_8487_ATP_Olambda__303,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_gw(fun(A,B),fun(A,B),Uu),Uua) = exp(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_303
tff(fact_8488_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qm(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_304
tff(fact_8489_ATP_Olambda__305,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_aci(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_305
tff(fact_8490_ATP_Olambda__306,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_adt(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_306
tff(fact_8491_ATP_Olambda__307,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_adx(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_307
tff(fact_8492_ATP_Olambda__308,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_vv(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_308
tff(fact_8493_ATP_Olambda__309,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_vw(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_309
tff(fact_8494_ATP_Olambda__310,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_nf(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_310
tff(fact_8495_ATP_Olambda__311,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_rx(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_311
tff(fact_8496_ATP_Olambda__312,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_zg(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_312
tff(fact_8497_ATP_Olambda__313,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_313
tff(fact_8498_ATP_Olambda__314,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: A] : aa(A,set(B),aTP_Lamp_ahx(fun(A,fun(B,$o)),fun(A,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),Uu,Uua)) ).

% ATP.lambda_314
tff(fact_8499_ATP_Olambda__315,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_abk(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_315
tff(fact_8500_ATP_Olambda__316,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_js(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_316
tff(fact_8501_ATP_Olambda__317,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_nj(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_317
tff(fact_8502_ATP_Olambda__318,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aji(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Uu)) ).

% ATP.lambda_318
tff(fact_8503_ATP_Olambda__319,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_acn(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_acm(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua))) ).

% ATP.lambda_319
tff(fact_8504_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real] : aa(real,filter(A),aTP_Lamp_wg(A,fun(real,filter(A)),Uu),Uua) = principal(A,aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_wf(A,fun(real,fun(A,$o)),Uu),Uua))) ) ).

% ATP.lambda_320
tff(fact_8505_ATP_Olambda__321,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aam(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I4: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_321
tff(fact_8506_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aar(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X4: set(A)] :
                  ( member(set(A),X4,Uu)
                 => member(A,aa(set(A),A,F7,X4),X4) ) ) ) ) ).

% ATP.lambda_322
tff(fact_8507_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aas(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X4: set(A)] :
                  ( member(set(A),X4,Uu)
                 => member(A,aa(set(A),A,F7,X4),X4) ) ) ) ) ).

% ATP.lambda_323
tff(fact_8508_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aaq(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X4: set(A)] :
                  ( member(set(A),X4,Uu)
                 => member(A,aa(set(A),A,F7,X4),X4) ) ) ) ) ).

% ATP.lambda_324
tff(fact_8509_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aan(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( member(A,X4,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X4) ) ) ) ).

% ATP.lambda_325
tff(fact_8510_ATP_Olambda__326,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aak(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( member(A,X4,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X4) ) ) ) ).

% ATP.lambda_326
tff(fact_8511_ATP_Olambda__327,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aao(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( member(A,X4,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Uua) ) ) ) ).

% ATP.lambda_327
tff(fact_8512_ATP_Olambda__328,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aal(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X4: A] :
              ( member(A,X4,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Uua) ) ) ) ).

% ATP.lambda_328
tff(fact_8513_ATP_Olambda__329,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_vb(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y4)
             => aa(A,$o,Uu,Y4) ) ) ) ).

% ATP.lambda_329
tff(fact_8514_ATP_Olambda__330,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aks(set(product_prod(A,A)),fun(set(A),$o),Uu),Uua)
    <=> ! [X4: A] :
          ( member(A,X4,Uua)
         => ! [Xa2: A] :
              ( member(A,Xa2,Uua)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),Uu)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),Uu) ) ) ) ) ).

% ATP.lambda_330
tff(fact_8515_ATP_Olambda__331,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ach(fun(B,option(A)),fun(A,$o),Uu),Uua)
    <=> ? [A7: B] : aa(B,option(A),Uu,A7) = aa(A,option(A),some(A),Uua) ) ).

% ATP.lambda_331
tff(fact_8516_ATP_Olambda__332,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_acf(fun(A,option(B)),fun(product_prod(A,B),$o),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_332
tff(fact_8517_ATP_Olambda__333,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( aa(product_prod(set(A),set(A)),$o,aTP_Lamp_ajx(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu),Uua)
    <=> ? [X9: 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)),X9),Y7) )
          & ( X9 != bot_bot(set(A)) )
          & ! [X4: A] :
              ( member(A,X4,Y7)
             => ? [Xa2: A] :
                  ( member(A,Xa2,X9)
                  & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),Uu) ) ) ) ) ).

% ATP.lambda_333
tff(fact_8518_ATP_Olambda__334,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_ahs(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_334
tff(fact_8519_ATP_Olambda__335,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_ahj(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_335
tff(fact_8520_ATP_Olambda__336,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_ahu(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_336
tff(fact_8521_ATP_Olambda__337,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,set(B),aTP_Lamp_ahp(fun(B,$o),fun(A,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),Uu) ).

% ATP.lambda_337
tff(fact_8522_ATP_Olambda__338,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_akb(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_338
tff(fact_8523_ATP_Olambda__339,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_dm(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_339
tff(fact_8524_ATP_Olambda__340,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bh(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_340
tff(fact_8525_ATP_Olambda__341,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_bc(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_341
tff(fact_8526_ATP_Olambda__342,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aeu(A,fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = $ite(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_342
tff(fact_8527_ATP_Olambda__343,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_ey(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(member(nat,Uub,Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_343
tff(fact_8528_ATP_Olambda__344,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_kr(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_344
tff(fact_8529_ATP_Olambda__345,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_ks(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_345
tff(fact_8530_ATP_Olambda__346,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bp(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_346
tff(fact_8531_ATP_Olambda__347,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_dv(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_347
tff(fact_8532_ATP_Olambda__348,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_ep(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_348
tff(fact_8533_ATP_Olambda__349,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bo(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_349
tff(fact_8534_ATP_Olambda__350,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_dw(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_350
tff(fact_8535_ATP_Olambda__351,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_abl(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),Uub),Uub) ).

% ATP.lambda_351
tff(fact_8536_ATP_Olambda__352,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ex(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(aa(nat,$o,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_352
tff(fact_8537_ATP_Olambda__353,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
          aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_agh(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_353
tff(fact_8538_ATP_Olambda__354,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_by(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_354
tff(fact_8539_ATP_Olambda__355,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_eb(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_355
tff(fact_8540_ATP_Olambda__356,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: fun(B,$o),Uub: B] :
      aa(B,fun(A,A),aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_agz(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,fun(A,A),Uu,Uub),id(A)) ).

% ATP.lambda_356
tff(fact_8541_ATP_Olambda__357,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_agv(fun(B,A),fun(fun(B,$o),fun(B,option(A))),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_357
tff(fact_8542_ATP_Olambda__358,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_aiw(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_aiv(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_358
tff(fact_8543_ATP_Olambda__359,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_abx(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_abn(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_359
tff(fact_8544_ATP_Olambda__360,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_adk(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_360
tff(fact_8545_ATP_Olambda__361,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_zf(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uub),Uu),Uua) ).

% ATP.lambda_361
tff(fact_8546_ATP_Olambda__362,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_kf(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),Uu),Uua),Uub) = aa(B,C,aa(D,fun(B,C),Uu,aa(A,D,Uua,aa(product_prod(A,B),A,product_fst(A,B),Uub))),aa(product_prod(A,B),B,product_snd(A,B),Uub)) ).

% ATP.lambda_362
tff(fact_8547_ATP_Olambda__363,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_363
tff(fact_8548_ATP_Olambda__364,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_364
tff(fact_8549_ATP_Olambda__365,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_fm(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_365
tff(fact_8550_ATP_Olambda__366,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_fj(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_366
tff(fact_8551_ATP_Olambda__367,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_aja(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ).

% ATP.lambda_367
tff(fact_8552_ATP_Olambda__368,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_zx(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_368
tff(fact_8553_ATP_Olambda__369,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_rc(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_369
tff(fact_8554_ATP_Olambda__370,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_ahb(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_370
tff(fact_8555_ATP_Olambda__371,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_wm(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_371
tff(fact_8556_ATP_Olambda__372,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_pv(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_372
tff(fact_8557_ATP_Olambda__373,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_dy(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_373
tff(fact_8558_ATP_Olambda__374,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_bv(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_374
tff(fact_8559_ATP_Olambda__375,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_aha(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_375
tff(fact_8560_ATP_Olambda__376,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agp(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_376
tff(fact_8561_ATP_Olambda__377,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_ii(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_ih(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_377
tff(fact_8562_ATP_Olambda__378,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_ig(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_if(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_378
tff(fact_8563_ATP_Olambda__379,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_id(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_ic(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_379
tff(fact_8564_ATP_Olambda__380,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_ez(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu,one_one(A),zero_zero(A))),
            aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_380
tff(fact_8565_ATP_Olambda__381,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_fn(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_fm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_381
tff(fact_8566_ATP_Olambda__382,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_fk(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_fj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_382
tff(fact_8567_ATP_Olambda__383,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_pw(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_pv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_383
tff(fact_8568_ATP_Olambda__384,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wn(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_wm(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_384
tff(fact_8569_ATP_Olambda__385,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_op(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uub),zero_zero(real)),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_385
tff(fact_8570_ATP_Olambda__386,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_oq(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real)),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_386
tff(fact_8571_ATP_Olambda__387,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_ho(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A)),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ) ).

% ATP.lambda_387
tff(fact_8572_ATP_Olambda__388,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_fy(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_388
tff(fact_8573_ATP_Olambda__389,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_rr(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_389
tff(fact_8574_ATP_Olambda__390,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_ro(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_390
tff(fact_8575_ATP_Olambda__391,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_ce(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,zero_zero(A)),Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_391
tff(fact_8576_ATP_Olambda__392,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_rp(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_392
tff(fact_8577_ATP_Olambda__393,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_om(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_393
tff(fact_8578_ATP_Olambda__394,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_hp(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,Uua,Uub),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_394
tff(fact_8579_ATP_Olambda__395,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_afs(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_395
tff(fact_8580_ATP_Olambda__396,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_ht(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu,Uub))),aa(nat,A,power_power(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_396
tff(fact_8581_ATP_Olambda__397,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ada(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub),lex(A,Uu)) ) ) ) ).

% ATP.lambda_397
tff(fact_8582_ATP_Olambda__398,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_afl(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).

% ATP.lambda_398
tff(fact_8583_ATP_Olambda__399,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_aeq(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X4: 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),X4),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)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4),Uu) ) ) ) ).

% ATP.lambda_399
tff(fact_8584_ATP_Olambda__400,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_aft(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_400
tff(fact_8585_ATP_Olambda__401,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_jw(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_401
tff(fact_8586_ATP_Olambda__402,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_jv(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_402
tff(fact_8587_ATP_Olambda__403,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ahm(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_403
tff(fact_8588_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_kc(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_404
tff(fact_8589_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_av(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_405
tff(fact_8590_ATP_Olambda__406,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_au(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_406
tff(fact_8591_ATP_Olambda__407,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_afr(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_407
tff(fact_8592_ATP_Olambda__408,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,option(B))] :
      ( aa(fun(A,option(B)),$o,aa(set(B),fun(fun(A,option(B)),$o),aTP_Lamp_acj(set(A),fun(set(B),fun(fun(A,option(B)),$o)),Uu),Uua),Uub)
    <=> ( ( dom(A,B,Uub) = Uu )
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),ran(A,B,Uub)),Uua) ) ) ).

% ATP.lambda_408
tff(fact_8593_ATP_Olambda__409,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_oi(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_409
tff(fact_8594_ATP_Olambda__410,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jq(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_410
tff(fact_8595_ATP_Olambda__411,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_agu(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Uu),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua),Uu) ) ).

% ATP.lambda_411
tff(fact_8596_ATP_Olambda__412,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_kd(nat,fun(nat,fun(set(nat),$o)),Uu),Uua),Uub)
    <=> ( member(set(nat),Uub,pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_412
tff(fact_8597_ATP_Olambda__413,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_acd(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))
        & member(nat,Uub,Uua) ) ) ).

% ATP.lambda_413
tff(fact_8598_ATP_Olambda__414,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_agm(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua))
        & aa(A,$o,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_414
tff(fact_8599_ATP_Olambda__415,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahi(list(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uu))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_415
tff(fact_8600_ATP_Olambda__416,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_afy(fun(A,$o),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uua))
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_416
tff(fact_8601_ATP_Olambda__417,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ji(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_417
tff(fact_8602_ATP_Olambda__418,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aki(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu))
        & ! [X4: A] :
            ( member(A,X4,Uua)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4),Uu) ) ) ) ).

% ATP.lambda_418
tff(fact_8603_ATP_Olambda__419,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_akk(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu))
        & ! [X4: A] :
            ( member(A,X4,Uua)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub),Uu) ) ) ) ).

% ATP.lambda_419
tff(fact_8604_ATP_Olambda__420,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_akj(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu))
        & ! [X4: A] :
            ( member(A,X4,Uua)
           => ( ( Uub != X4 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4),Uu) ) ) ) ) ).

% ATP.lambda_420
tff(fact_8605_ATP_Olambda__421,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_aem(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)),Uua) ) ).

% ATP.lambda_421
tff(fact_8606_ATP_Olambda__422,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ajj(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uub) ) ) ).

% ATP.lambda_422
tff(fact_8607_ATP_Olambda__423,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_all(set(A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,Uu),Uua)),Uub) ) ) ) ).

% ATP.lambda_423
tff(fact_8608_ATP_Olambda__424,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajq(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_424
tff(fact_8609_ATP_Olambda__425,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_ajw(set(nat),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & member(nat,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ajv(set(nat),fun(nat,fun(nat,$o)),Uu),Uub))),Uua) ) ) ).

% ATP.lambda_425
tff(fact_8610_ATP_Olambda__426,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_jo(set(A),fun(nat,fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu)
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_426
tff(fact_8611_ATP_Olambda__427,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jr(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_427
tff(fact_8612_ATP_Olambda__428,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_hs(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_428
tff(fact_8613_ATP_Olambda__429,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_of(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_429
tff(fact_8614_ATP_Olambda__430,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_ia(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,power_power(A,Uu),Uub)) ) ).

% ATP.lambda_430
tff(fact_8615_ATP_Olambda__431,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A)),Uub: fun(B,A)] : aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_adb(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),Uu),Uua),Uub) = aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),inv_image(A,B,Uu,Uub)),inv_image(A,B,Uua,Uub)) ).

% ATP.lambda_431
tff(fact_8616_ATP_Olambda__432,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fw(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).

% ATP.lambda_432
tff(fact_8617_ATP_Olambda__433,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bj(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_433
tff(fact_8618_ATP_Olambda__434,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ac(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_434
tff(fact_8619_ATP_Olambda__435,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bk(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_435
tff(fact_8620_ATP_Olambda__436,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ag(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_436
tff(fact_8621_ATP_Olambda__437,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_af(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_437
tff(fact_8622_ATP_Olambda__438,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ad(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_438
tff(fact_8623_ATP_Olambda__439,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_akr(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu) ) ) ).

% ATP.lambda_439
tff(fact_8624_ATP_Olambda__440,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_rw(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uu) ) ) ).

% ATP.lambda_440
tff(fact_8625_ATP_Olambda__441,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajf(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uua,Uu)
        & member(A,Uub,Uu) ) ) ).

% ATP.lambda_441
tff(fact_8626_ATP_Olambda__442,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ajv(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_442
tff(fact_8627_ATP_Olambda__443,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_wz(set(A),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_443
tff(fact_8628_ATP_Olambda__444,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_afm(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_444
tff(fact_8629_ATP_Olambda__445,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_afu(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_445
tff(fact_8630_ATP_Olambda__446,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bx(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_446
tff(fact_8631_ATP_Olambda__447,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cg(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_447
tff(fact_8632_ATP_Olambda__448,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cf(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_448
tff(fact_8633_ATP_Olambda__449,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( aa(A,$o,aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_aif(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_449
tff(fact_8634_ATP_Olambda__450,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_450
tff(fact_8635_ATP_Olambda__451,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_iw(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_451
tff(fact_8636_ATP_Olambda__452,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_452
tff(fact_8637_ATP_Olambda__453,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ix(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_453
tff(fact_8638_ATP_Olambda__454,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_mi(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_454
tff(fact_8639_ATP_Olambda__455,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_jn(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_455
tff(fact_8640_ATP_Olambda__456,axiom,
    ! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ady(list(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_456
tff(fact_8641_ATP_Olambda__457,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ru(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_457
tff(fact_8642_ATP_Olambda__458,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_fr(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_458
tff(fact_8643_ATP_Olambda__459,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_ti(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_459
tff(fact_8644_ATP_Olambda__460,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_xj(real,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(complex,Uua,Uub)),Uu) ) ).

% ATP.lambda_460
tff(fact_8645_ATP_Olambda__461,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_xl(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(real,Uua,Uub)),Uu) ) ).

% ATP.lambda_461
tff(fact_8646_ATP_Olambda__462,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xg(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_462
tff(fact_8647_ATP_Olambda__463,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ng(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_463
tff(fact_8648_ATP_Olambda__464,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_wf(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_464
tff(fact_8649_ATP_Olambda__465,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ga(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_465
tff(fact_8650_ATP_Olambda__466,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mn(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_466
tff(fact_8651_ATP_Olambda__467,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ml(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_467
tff(fact_8652_ATP_Olambda__468,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mm(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_468
tff(fact_8653_ATP_Olambda__469,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mk(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_469
tff(fact_8654_ATP_Olambda__470,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alf(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu) ) ).

% ATP.lambda_470
tff(fact_8655_ATP_Olambda__471,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_aq(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu) ) ).

% ATP.lambda_471
tff(fact_8656_ATP_Olambda__472,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_yb(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ).

% ATP.lambda_472
tff(fact_8657_ATP_Olambda__473,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_akm(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu) ) ).

% ATP.lambda_473
tff(fact_8658_ATP_Olambda__474,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afh(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_474
tff(fact_8659_ATP_Olambda__475,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_aw(nat,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_475
tff(fact_8660_ATP_Olambda__476,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_jt(complex,fun(nat,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_476
tff(fact_8661_ATP_Olambda__477,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_zs(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_477
tff(fact_8662_ATP_Olambda__478,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_hl(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_478
tff(fact_8663_ATP_Olambda__479,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_on(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,power_power(real,Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_479
tff(fact_8664_ATP_Olambda__480,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_hm(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_480
tff(fact_8665_ATP_Olambda__481,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gg(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,power_power(nat,Uua),Uub)) ).

% ATP.lambda_481
tff(fact_8666_ATP_Olambda__482,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_vy(fun(nat,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,power_power(B,Uua),Uub)) ) ).

% ATP.lambda_482
tff(fact_8667_ATP_Olambda__483,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_fh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_483
tff(fact_8668_ATP_Olambda__484,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_484
tff(fact_8669_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hk(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_485
tff(fact_8670_ATP_Olambda__486,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_486
tff(fact_8671_ATP_Olambda__487,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_fq(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_487
tff(fact_8672_ATP_Olambda__488,axiom,
    ! [A: $tType] :
      ( idom(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,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_488
tff(fact_8673_ATP_Olambda__489,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ar(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_489
tff(fact_8674_ATP_Olambda__490,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pd(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_490
tff(fact_8675_ATP_Olambda__491,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_air(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uub)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_491
tff(fact_8676_ATP_Olambda__492,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_aiq(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uub)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_492
tff(fact_8677_ATP_Olambda__493,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_afz(fun(B,A),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_493
tff(fact_8678_ATP_Olambda__494,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_yx(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_494
tff(fact_8679_ATP_Olambda__495,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ty(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_495
tff(fact_8680_ATP_Olambda__496,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ud(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_496
tff(fact_8681_ATP_Olambda__497,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ul(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_497
tff(fact_8682_ATP_Olambda__498,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pt(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_498
tff(fact_8683_ATP_Olambda__499,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_pk(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_499
tff(fact_8684_ATP_Olambda__500,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_wt(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_500
tff(fact_8685_ATP_Olambda__501,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_nz(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_501
tff(fact_8686_ATP_Olambda__502,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_vn(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_502
tff(fact_8687_ATP_Olambda__503,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_uv(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_503
tff(fact_8688_ATP_Olambda__504,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_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_504
tff(fact_8689_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qs(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_505
tff(fact_8690_ATP_Olambda__506,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ww(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_506
tff(fact_8691_ATP_Olambda__507,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aky(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_507
tff(fact_8692_ATP_Olambda__508,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_jm(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_508
tff(fact_8693_ATP_Olambda__509,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_jl(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_509
tff(fact_8694_ATP_Olambda__510,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pi(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_510
tff(fact_8695_ATP_Olambda__511,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_nv(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_511
tff(fact_8696_ATP_Olambda__512,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_sz(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_512
tff(fact_8697_ATP_Olambda__513,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_513
tff(fact_8698_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tl(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_514
tff(fact_8699_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_515
tff(fact_8700_ATP_Olambda__516,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_ta(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_516
tff(fact_8701_ATP_Olambda__517,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_hy(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_517
tff(fact_8702_ATP_Olambda__518,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_aia(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_518
tff(fact_8703_ATP_Olambda__519,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_519
tff(fact_8704_ATP_Olambda__520,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_520
tff(fact_8705_ATP_Olambda__521,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_df(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_521
tff(fact_8706_ATP_Olambda__522,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ch(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_522
tff(fact_8707_ATP_Olambda__523,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_523
tff(fact_8708_ATP_Olambda__524,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_524
tff(fact_8709_ATP_Olambda__525,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vl(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_525
tff(fact_8710_ATP_Olambda__526,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_rb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8711_ATP_Olambda__527,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_aib(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_527
tff(fact_8712_ATP_Olambda__528,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_abb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8713_ATP_Olambda__529,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ahy(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_529
tff(fact_8714_ATP_Olambda__530,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_530
tff(fact_8715_ATP_Olambda__531,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8716_ATP_Olambda__532,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_bs(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_532
tff(fact_8717_ATP_Olambda__533,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ph(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_533
tff(fact_8718_ATP_Olambda__534,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_534
tff(fact_8719_ATP_Olambda__535,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_nw(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_535
tff(fact_8720_ATP_Olambda__536,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_536
tff(fact_8721_ATP_Olambda__537,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_537
tff(fact_8722_ATP_Olambda__538,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_538
tff(fact_8723_ATP_Olambda__539,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_iy(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_539
tff(fact_8724_ATP_Olambda__540,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ok(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_540
tff(fact_8725_ATP_Olambda__541,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_541
tff(fact_8726_ATP_Olambda__542,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xa(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_542
tff(fact_8727_ATP_Olambda__543,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_uu(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_543
tff(fact_8728_ATP_Olambda__544,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_xb(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_544
tff(fact_8729_ATP_Olambda__545,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_vt(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_545
tff(fact_8730_ATP_Olambda__546,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_rk(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_546
tff(fact_8731_ATP_Olambda__547,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_aix(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_547
tff(fact_8732_ATP_Olambda__548,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_yr(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_548
tff(fact_8733_ATP_Olambda__549,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_ws(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_549
tff(fact_8734_ATP_Olambda__550,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_vk(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_550
tff(fact_8735_ATP_Olambda__551,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_rf(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_551
tff(fact_8736_ATP_Olambda__552,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_xu(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_552
tff(fact_8737_ATP_Olambda__553,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,B)),Uua: fun(A,nat),Uub: A] : aa(A,fun(B,B),aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_abz(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),Uu),Uua),Uub) = aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(A,nat,Uua,Uub)),aa(A,fun(B,B),Uu,Uub)) ).

% ATP.lambda_553
tff(fact_8738_ATP_Olambda__554,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_554
tff(fact_8739_ATP_Olambda__555,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_555
tff(fact_8740_ATP_Olambda__556,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_556
tff(fact_8741_ATP_Olambda__557,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agg(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_557
tff(fact_8742_ATP_Olambda__558,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_agf(A,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_558
tff(fact_8743_ATP_Olambda__559,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ko(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_559
tff(fact_8744_ATP_Olambda__560,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zq(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_560
tff(fact_8745_ATP_Olambda__561,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_vh(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & ! [Y4: A] :
            ( aa(A,$o,Uua,Y4)
           => aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Y4) ) ) ) ).

% ATP.lambda_561
tff(fact_8746_ATP_Olambda__562,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_oc(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_562
tff(fact_8747_ATP_Olambda__563,axiom,
    ! [A: $tType,B: $tType] :
      ( order(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,set(B),aa(B,fun(A,set(B)),aTP_Lamp_aip(fun(A,B),fun(B,fun(A,set(B))),Uu),Uua),Uub) = set_or3652927894154168847AtMost(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_563
tff(fact_8748_ATP_Olambda__564,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_rd(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_564
tff(fact_8749_ATP_Olambda__565,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_as(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_565
tff(fact_8750_ATP_Olambda__566,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_un(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_566
tff(fact_8751_ATP_Olambda__567,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tw(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_567
tff(fact_8752_ATP_Olambda__568,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_568
tff(fact_8753_ATP_Olambda__569,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_569
tff(fact_8754_ATP_Olambda__570,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_he(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uu,Uub),Uua) ) ).

% ATP.lambda_570
tff(fact_8755_ATP_Olambda__571,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_kv(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_571
tff(fact_8756_ATP_Olambda__572,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qt(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_572
tff(fact_8757_ATP_Olambda__573,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_sb(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uua,Uub),Uu) ) ).

% ATP.lambda_573
tff(fact_8758_ATP_Olambda__574,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_uc(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_574
tff(fact_8759_ATP_Olambda__575,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_575
tff(fact_8760_ATP_Olambda__576,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tu(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_576
tff(fact_8761_ATP_Olambda__577,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wl(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_577
tff(fact_8762_ATP_Olambda__578,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qx(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_578
tff(fact_8763_ATP_Olambda__579,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_es(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_579
tff(fact_8764_ATP_Olambda__580,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_rz(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_580
tff(fact_8765_ATP_Olambda__581,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_qg(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_581
tff(fact_8766_ATP_Olambda__582,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_nl(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_582
tff(fact_8767_ATP_Olambda__583,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_qq(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_583
tff(fact_8768_ATP_Olambda__584,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_od(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_584
tff(fact_8769_ATP_Olambda__585,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_tn(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_585
tff(fact_8770_ATP_Olambda__586,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_ri(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_586
tff(fact_8771_ATP_Olambda__587,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_to(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_587
tff(fact_8772_ATP_Olambda__588,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qi(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_588
tff(fact_8773_ATP_Olambda__589,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_aba(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_589
tff(fact_8774_ATP_Olambda__590,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_mx(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_590
tff(fact_8775_ATP_Olambda__591,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_mg(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_591
tff(fact_8776_ATP_Olambda__592,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_uz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_592
tff(fact_8777_ATP_Olambda__593,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_rv(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_593
tff(fact_8778_ATP_Olambda__594,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_oj(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_594
tff(fact_8779_ATP_Olambda__595,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tb(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_595
tff(fact_8780_ATP_Olambda__596,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_xq(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_596
tff(fact_8781_ATP_Olambda__597,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: int,Uub: B] : aa(B,A,aa(int,fun(B,A),aTP_Lamp_zo(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_597
tff(fact_8782_ATP_Olambda__598,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_zm(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_598
tff(fact_8783_ATP_Olambda__599,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_zj(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_599
tff(fact_8784_ATP_Olambda__600,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_zn(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_600
tff(fact_8785_ATP_Olambda__601,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_zl(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_601
tff(fact_8786_ATP_Olambda__602,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_zk(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_602
tff(fact_8787_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(B,A),Uub: B] :
          ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_yy(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(A,aa(B,A,Uua,Uub),Uu) ) ) ).

% ATP.lambda_603
tff(fact_8788_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_yw(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu) ) ).

% ATP.lambda_604
tff(fact_8789_ATP_Olambda__605,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ake(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_605
tff(fact_8790_ATP_Olambda__606,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ajg(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_606
tff(fact_8791_ATP_Olambda__607,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_akl(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu) ) ) ).

% ATP.lambda_607
tff(fact_8792_ATP_Olambda__608,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajl(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu) ) ) ).

% ATP.lambda_608
tff(fact_8793_ATP_Olambda__609,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jh(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_609
tff(fact_8794_ATP_Olambda__610,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_hn(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ) ).

% ATP.lambda_610
tff(fact_8795_ATP_Olambda__611,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_kp(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_611
tff(fact_8796_ATP_Olambda__612,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_uy(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua) ) ) ).

% ATP.lambda_612
tff(fact_8797_ATP_Olambda__613,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_io(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)))),aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_613
tff(fact_8798_ATP_Olambda__614,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zr(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_614
tff(fact_8799_ATP_Olambda__615,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aaw(fun(A,A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( ? [X4: A] :
                ( ( Uub = aa(A,A,Uu,X4) )
                & aa(A,$o,Uua,X4) )
            | ? [M8: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X4: A] :
                    ( member(A,X4,M8)
                   => aa(A,$o,Uua,X4) ) ) ) ) ) ).

% ATP.lambda_615
tff(fact_8800_ATP_Olambda__616,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),$o)),Uua: list(A),Uub: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_aej(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub)
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4) )
            | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X4)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu,Xs3),Ys4) ) ) ) ) ).

% ATP.lambda_616
tff(fact_8801_ATP_Olambda__617,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_acm(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,finite_finite2(A),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu) ) ) ).

% ATP.lambda_617
tff(fact_8802_ATP_Olambda__618,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_ajr(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,finite_finite2(A),Uua)
        & aa(set(A),$o,finite_finite2(A),Uub)
        & ( Uub != bot_bot(set(A)) )
        & ! [X4: A] :
            ( member(A,X4,Uua)
           => ? [Xa2: A] :
                ( member(A,Xa2,Uub)
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),Uu) ) ) ) ) ).

% ATP.lambda_618
tff(fact_8803_ATP_Olambda__619,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_619
tff(fact_8804_ATP_Olambda__620,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jg(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,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_620
tff(fact_8805_ATP_Olambda__621,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_je(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,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_621
tff(fact_8806_ATP_Olambda__622,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_zi(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_622
tff(fact_8807_ATP_Olambda__623,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ct(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_623
tff(fact_8808_ATP_Olambda__624,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cw(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_624
tff(fact_8809_ATP_Olambda__625,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_abi(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_625
tff(fact_8810_ATP_Olambda__626,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dl(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_626
tff(fact_8811_ATP_Olambda__627,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sm(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = divide_divide(real,Uua,aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_627
tff(fact_8812_ATP_Olambda__628,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ck(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_628
tff(fact_8813_ATP_Olambda__629,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_cb(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_629
tff(fact_8814_ATP_Olambda__630,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_adw(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_630
tff(fact_8815_ATP_Olambda__631,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_adu(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_631
tff(fact_8816_ATP_Olambda__632,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_aeh(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),take(A,Uub,Uua)) ).

% ATP.lambda_632
tff(fact_8817_ATP_Olambda__633,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_agq(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_633
tff(fact_8818_ATP_Olambda__634,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_ug(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu,Uua)) ) ).

% ATP.lambda_634
tff(fact_8819_ATP_Olambda__635,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ne(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_635
tff(fact_8820_ATP_Olambda__636,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_nd(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_636
tff(fact_8821_ATP_Olambda__637,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ui(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_637
tff(fact_8822_ATP_Olambda__638,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_um(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_638
tff(fact_8823_ATP_Olambda__639,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uk(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_639
tff(fact_8824_ATP_Olambda__640,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ue(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_640
tff(fact_8825_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ub(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_641
tff(fact_8826_ATP_Olambda__642,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ahg(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_642
tff(fact_8827_ATP_Olambda__643,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uf(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_643
tff(fact_8828_ATP_Olambda__644,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_644
tff(fact_8829_ATP_Olambda__645,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wk(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_645
tff(fact_8830_ATP_Olambda__646,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_et(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_646
tff(fact_8831_ATP_Olambda__647,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_er(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_647
tff(fact_8832_ATP_Olambda__648,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_sa(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_648
tff(fact_8833_ATP_Olambda__649,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_qh(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_649
tff(fact_8834_ATP_Olambda__650,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_qy(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_650
tff(fact_8835_ATP_Olambda__651,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_we(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_651
tff(fact_8836_ATP_Olambda__652,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_na(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_652
tff(fact_8837_ATP_Olambda__653,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_sq(fun(A,nat),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,Uua),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_653
tff(fact_8838_ATP_Olambda__654,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_aaz(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_654
tff(fact_8839_ATP_Olambda__655,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_my(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_655
tff(fact_8840_ATP_Olambda__656,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mf(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_656
tff(fact_8841_ATP_Olambda__657,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qv(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_657
tff(fact_8842_ATP_Olambda__658,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_kw(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_658
tff(fact_8843_ATP_Olambda__659,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_xr(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_659
tff(fact_8844_ATP_Olambda__660,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_ads(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_660
tff(fact_8845_ATP_Olambda__661,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_mt(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)),insert(A),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_661
tff(fact_8846_ATP_Olambda__662,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_xd(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),Uu),Uua),Uub) = aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_662
tff(fact_8847_ATP_Olambda__663,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_aju(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        | aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_663
tff(fact_8848_ATP_Olambda__664,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_abo(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_664
tff(fact_8849_ATP_Olambda__665,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_age(fun(list(A),A),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_665
tff(fact_8850_ATP_Olambda__666,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_zd(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_666
tff(fact_8851_ATP_Olambda__667,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ej(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_667
tff(fact_8852_ATP_Olambda__668,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_agi(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(list(A),set(A),set2(A),nths(A,Uu,Uua))) ) ).

% ATP.lambda_668
tff(fact_8853_ATP_Olambda__669,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_us(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_669
tff(fact_8854_ATP_Olambda__670,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_ahz(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)),Uu),Uua) ).

% ATP.lambda_670
tff(fact_8855_ATP_Olambda__671,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_agx(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_671
tff(fact_8856_ATP_Olambda__672,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_ahv(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uu),Uua) ).

% ATP.lambda_672
tff(fact_8857_ATP_Olambda__673,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_ain(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_673
tff(fact_8858_ATP_Olambda__674,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ef(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_674
tff(fact_8859_ATP_Olambda__675,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_675
tff(fact_8860_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sg(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_676
tff(fact_8861_ATP_Olambda__677,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_rh(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_677
tff(fact_8862_ATP_Olambda__678,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_nu(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_678
tff(fact_8863_ATP_Olambda__679,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_tq(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_679
tff(fact_8864_ATP_Olambda__680,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_nb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_680
tff(fact_8865_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fa(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_681
tff(fact_8866_ATP_Olambda__682,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_682
tff(fact_8867_ATP_Olambda__683,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_va(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_683
tff(fact_8868_ATP_Olambda__684,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ed(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_684
tff(fact_8869_ATP_Olambda__685,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cj(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_685
tff(fact_8870_ATP_Olambda__686,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_uh(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_686
tff(fact_8871_ATP_Olambda__687,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_rj(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_687
tff(fact_8872_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_rt(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_8873_ATP_Olambda__689,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ny(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_689
tff(fact_8874_ATP_Olambda__690,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ajt(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(A,B),$o,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_8875_ATP_Olambda__691,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_bg(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_691
tff(fact_8876_ATP_Olambda__692,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xi(fun(product_prod(A,A),$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(product_prod(A,A),$o,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)) ) ) ).

% ATP.lambda_692
tff(fact_8877_ATP_Olambda__693,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_rn(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),Uub)) ) ).

% ATP.lambda_693
tff(fact_8878_ATP_Olambda__694,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_vu(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_694
tff(fact_8879_ATP_Olambda__695,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_ew(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_695
tff(fact_8880_ATP_Olambda__696,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_acy(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_696
tff(fact_8881_ATP_Olambda__697,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: A] : aa(A,fun(B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_kg(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),Uu),Uua),Uub) = aa(D,fun(B,C),Uu,aa(A,D,Uua,Uub)) ).

% ATP.lambda_697
tff(fact_8882_ATP_Olambda__698,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_jx(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_698
tff(fact_8883_ATP_Olambda__699,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_aad(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_699
tff(fact_8884_ATP_Olambda__700,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_akd(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_700
tff(fact_8885_ATP_Olambda__701,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_ye(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_701
tff(fact_8886_ATP_Olambda__702,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_wj(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_702
tff(fact_8887_ATP_Olambda__703,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wp(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_703
tff(fact_8888_ATP_Olambda__704,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_sy(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_704
tff(fact_8889_ATP_Olambda__705,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_ald(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_705
tff(fact_8890_ATP_Olambda__706,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_yc(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_8891_ATP_Olambda__707,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,$o),aa(fun(C,A),fun(C,fun(B,$o)),aTP_Lamp_akv(fun(A,fun(B,$o)),fun(fun(C,A),fun(C,fun(B,$o))),Uu),Uua),Uub) = aa(A,fun(B,$o),Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_707
tff(fact_8892_ATP_Olambda__708,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_acw(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_708
tff(fact_8893_ATP_Olambda__709,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_acx(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_709
tff(fact_8894_ATP_Olambda__710,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_nm(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_710
tff(fact_8895_ATP_Olambda__711,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_rq(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_711
tff(fact_8896_ATP_Olambda__712,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_rm(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_712
tff(fact_8897_ATP_Olambda__713,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ajo(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_713
tff(fact_8898_ATP_Olambda__714,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ajn(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_714
tff(fact_8899_ATP_Olambda__715,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_mo(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_715
tff(fact_8900_ATP_Olambda__716,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_aig(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).

% ATP.lambda_716
tff(fact_8901_ATP_Olambda__717,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xt(fun(B,$o),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_717
tff(fact_8902_ATP_Olambda__718,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_akp(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ako(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ).

% ATP.lambda_718
tff(fact_8903_ATP_Olambda__719,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_xx(fun(product_prod(A,A),$o),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_xw(fun(product_prod(A,A),$o),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ) ).

% ATP.lambda_719
tff(fact_8904_ATP_Olambda__720,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_wa(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_vz(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_720
tff(fact_8905_ATP_Olambda__721,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_qc(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_qb(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_721
tff(fact_8906_ATP_Olambda__722,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_wc(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)),insert(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_722
tff(fact_8907_ATP_Olambda__723,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_wb(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_lessThan(A),Uub)),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_723
tff(fact_8908_ATP_Olambda__724,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_vi(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)),insert(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_724
tff(fact_8909_ATP_Olambda__725,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub))) ) ).

% ATP.lambda_725
tff(fact_8910_ATP_Olambda__726,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aca(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_726
tff(fact_8911_ATP_Olambda__727,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_acb(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_727
tff(fact_8912_ATP_Olambda__728,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xf(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_728
tff(fact_8913_ATP_Olambda__729,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xe(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_729
tff(fact_8914_ATP_Olambda__730,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_kx(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_730
tff(fact_8915_ATP_Olambda__731,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_aij(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),Uu),Uua)) ).

% ATP.lambda_731
tff(fact_8916_ATP_Olambda__732,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_jp(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_732
tff(fact_8917_ATP_Olambda__733,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_adi(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [I4: nat] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu),I4)),aa(nat,B,nth(B,Uua),I4)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_733
tff(fact_8918_ATP_Olambda__734,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_acc(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [I4: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu))
          & member(nat,I4,Uua) ) ) ).

% ATP.lambda_734
tff(fact_8919_ATP_Olambda__735,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acu(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A7) )
              & member(A,A7,Uu) ) ) ) ).

% ATP.lambda_735
tff(fact_8920_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acs(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A7) )
              & member(A,A7,Uu) ) ) ) ).

% ATP.lambda_736
tff(fact_8921_ATP_Olambda__737,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,B),Uub: B] :
      ( aa(B,$o,aa(fun(A,B),fun(B,$o),aTP_Lamp_aab(fun(A,$o),fun(fun(A,B),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X4: A] :
          ( ( Uub = aa(A,B,Uua,X4) )
          & aa(A,$o,Uu,X4) ) ) ).

% ATP.lambda_737
tff(fact_8922_ATP_Olambda__738,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aaj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X4: A] :
          ( member(A,X4,Uu)
         => aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X4) ) ) ).

% ATP.lambda_738
tff(fact_8923_ATP_Olambda__739,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_ajs(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X4: A] :
          ( member(A,X4,Uu)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X4) ) ) ).

% ATP.lambda_739
tff(fact_8924_ATP_Olambda__740,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_alg(set(product_prod(B,A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( member(B,X4,Uua)
          & member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),Uub),Uu) ) ) ).

% ATP.lambda_740
tff(fact_8925_ATP_Olambda__741,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ajy(fun(B,option(A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X4: B] :
          ( member(B,X4,Uua)
          & ( aa(B,option(A),Uu,X4) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_741
tff(fact_8926_ATP_Olambda__742,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_uw(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,Uub,N)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_742
tff(fact_8927_ATP_Olambda__743,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_vd(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [A7: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A7)
             => ! [B7: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A7),B7)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or3652927894154168847AtMost(nat,A7,B7)))),aa(nat,real,Uua,A7)) ) ) ) ) ).

% ATP.lambda_743
tff(fact_8928_ATP_Olambda__744,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zw(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [Y4: A] :
          ( aa(A,$o,Uu,Y4)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),Y4) ) ) ).

% ATP.lambda_744
tff(fact_8929_ATP_Olambda__745,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aap(fun(A,A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ? [N: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),Uu),Uua) ) ).

% ATP.lambda_745
tff(fact_8930_ATP_Olambda__746,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_aep(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [A7: A,V7: 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),V7)) )
          | ? [U6: list(A),Aa3: A,B7: A,Va4: list(A),W3: list(A)] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),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),W3)) ) ) ) ) ).

% ATP.lambda_746
tff(fact_8931_ATP_Olambda__747,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_acv(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A7: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A7),B7) )
              & member(A,A7,Uu)
              & member(A,B7,Uua) ) ) ) ).

% ATP.lambda_747
tff(fact_8932_ATP_Olambda__748,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_act(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A7: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A7),B7) )
              & member(A,A7,Uu)
              & member(A,B7,Uua) ) ) ) ).

% ATP.lambda_748
tff(fact_8933_ATP_Olambda__749,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_ael(set(A),fun(set(list(A)),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [X4: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
          & member(A,X4,Uu)
          & member(list(A),Xs3,Uua) ) ) ).

% ATP.lambda_749
tff(fact_8934_ATP_Olambda__750,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_aeo(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [Us2: list(A),Z5: A,Z8: 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),Z5),Vs2)) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z5),Z8),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),Z8),Vs2)) ) ) ) ).

% ATP.lambda_750
tff(fact_8935_ATP_Olambda__751,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_ih(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc)))),semiring_char_0_fact(real,Uub))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_751
tff(fact_8936_ATP_Olambda__752,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_ic(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc)))),semiring_char_0_fact(real,Uub)))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_752
tff(fact_8937_ATP_Olambda__753,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_if(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc)))),semiring_char_0_fact(real,Uub))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_753
tff(fact_8938_ATP_Olambda__754,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_abr(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu) = Uua,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_754
tff(fact_8939_ATP_Olambda__755,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_abh(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image2(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_755
tff(fact_8940_ATP_Olambda__756,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_alj(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uua,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uub),Uuc),Uuc) ).

% ATP.lambda_756
tff(fact_8941_ATP_Olambda__757,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_gj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_757
tff(fact_8942_ATP_Olambda__758,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_gl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_758
tff(fact_8943_ATP_Olambda__759,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_wi(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_759
tff(fact_8944_ATP_Olambda__760,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_gm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_760
tff(fact_8945_ATP_Olambda__761,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_gk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_761
tff(fact_8946_ATP_Olambda__762,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_abc(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_762
tff(fact_8947_ATP_Olambda__763,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_fc(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_763
tff(fact_8948_ATP_Olambda__764,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_eh(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_764
tff(fact_8949_ATP_Olambda__765,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_cp(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_765
tff(fact_8950_ATP_Olambda__766,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_ju(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_766
tff(fact_8951_ATP_Olambda__767,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: B,Uub: B,Uuc: A] :
      aa(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_ajc(set(A),fun(B,fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uu),Uua,Uub) ).

% ATP.lambda_767
tff(fact_8952_ATP_Olambda__768,axiom,
    ! [A: $tType,Uu: fun(A,$o),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_ahl(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,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_768
tff(fact_8953_ATP_Olambda__769,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ku(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_769
tff(fact_8954_ATP_Olambda__770,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_kt(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_770
tff(fact_8955_ATP_Olambda__771,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_abw(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_abv(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_771
tff(fact_8956_ATP_Olambda__772,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_abu(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_abt(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_772
tff(fact_8957_ATP_Olambda__773,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_kk(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_773
tff(fact_8958_ATP_Olambda__774,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_agb(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ).

% ATP.lambda_774
tff(fact_8959_ATP_Olambda__775,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_ado(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_775
tff(fact_8960_ATP_Olambda__776,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_aat(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_776
tff(fact_8961_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(C,B),Uub: A,Uuc: C] :
      ( aa(C,$o,aa(A,fun(C,$o),aa(fun(C,B),fun(A,fun(C,$o)),aTP_Lamp_aku(fun(A,fun(B,$o)),fun(fun(C,B),fun(A,fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_777
tff(fact_8962_ATP_Olambda__778,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_adn(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_778
tff(fact_8963_ATP_Olambda__779,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_da(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_cz(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),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_779
tff(fact_8964_ATP_Olambda__780,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_gn(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(
                Uuc = zero_zero(nat),
                aa(A,A,uminus_uminus(A),Uub),
                $ite(Uuc = Uu,one_one(A),zero_zero(A)) )),
            aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_780
tff(fact_8965_ATP_Olambda__781,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_dz(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_dy(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_am(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_781
tff(fact_8966_ATP_Olambda__782,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_bw(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aTP_Lamp_bv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_am(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_782
tff(fact_8967_ATP_Olambda__783,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_gi(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_gh(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,nat,power_power(nat,Uub),Uuc)) ).

% ATP.lambda_783
tff(fact_8968_ATP_Olambda__784,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_gd(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_gc(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_784
tff(fact_8969_ATP_Olambda__785,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_ow(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real)),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,Uub),Uuc)) ).

% ATP.lambda_785
tff(fact_8970_ATP_Olambda__786,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_ou(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real)),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,Uub),Uuc)) ).

% ATP.lambda_786
tff(fact_8971_ATP_Olambda__787,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_os(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uua),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Uub),Uua)),Uuc)) ).

% ATP.lambda_787
tff(fact_8972_ATP_Olambda__788,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_or(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uub),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uub)),Uuc)) ).

% ATP.lambda_788
tff(fact_8973_ATP_Olambda__789,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dh(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uu),Uub)) ) ).

% ATP.lambda_789
tff(fact_8974_ATP_Olambda__790,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_ky(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_790
tff(fact_8975_ATP_Olambda__791,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_cz(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,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),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_791
tff(fact_8976_ATP_Olambda__792,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_sr(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_792
tff(fact_8977_ATP_Olambda__793,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_nr(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uua) ) ) ) ).

% ATP.lambda_793
tff(fact_8978_ATP_Olambda__794,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_adc(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc)),Uu) ) ).

% ATP.lambda_794
tff(fact_8979_ATP_Olambda__795,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ux(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_795
tff(fact_8980_ATP_Olambda__796,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_up(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_796
tff(fact_8981_ATP_Olambda__797,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gf(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_797
tff(fact_8982_ATP_Olambda__798,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_gb(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,power_power(nat,Uu),Uuc))),aa(nat,nat,power_power(nat,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_798
tff(fact_8983_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ge(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_799
tff(fact_8984_ATP_Olambda__800,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_il(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))),aa(nat,A,power_power(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_800
tff(fact_8985_ATP_Olambda__801,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_aes(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X4: 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),X4),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)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y4),Uu) ) ) ) ).

% ATP.lambda_801
tff(fact_8986_ATP_Olambda__802,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dj(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,power_power(A,Uu),Uuc)) ) ).

% ATP.lambda_802
tff(fact_8987_ATP_Olambda__803,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_803
tff(fact_8988_ATP_Olambda__804,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_am(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_804
tff(fact_8989_ATP_Olambda__805,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_di(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu),Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_805
tff(fact_8990_ATP_Olambda__806,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: fun(A,real),Uuc: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_wx(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uua,Uuc)),aa(A,real,Uub,Uuc)) ) ) ) ).

% ATP.lambda_806
tff(fact_8991_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_lv(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_807
tff(fact_8992_ATP_Olambda__808,axiom,
    ! [A: $tType,C: $tType,Uu: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ly(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_808
tff(fact_8993_ATP_Olambda__809,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_md(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_809
tff(fact_8994_ATP_Olambda__810,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_ajh(fun(A,B),fun(set(A),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uua)
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_810
tff(fact_8995_ATP_Olambda__811,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_yv(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uub)
        & member(A,aa(B,A,Uua,Uuc),Uu) ) ) ).

% ATP.lambda_811
tff(fact_8996_ATP_Olambda__812,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dk(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu),Uuc)),aa(nat,A,power_power(A,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).

% ATP.lambda_812
tff(fact_8997_ATP_Olambda__813,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fx(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_813
tff(fact_8998_ATP_Olambda__814,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_le(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_814
tff(fact_8999_ATP_Olambda__815,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_lc(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_815
tff(fact_9000_ATP_Olambda__816,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_lg(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_816
tff(fact_9001_ATP_Olambda__817,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_li(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_817
tff(fact_9002_ATP_Olambda__818,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_afa(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_818
tff(fact_9003_ATP_Olambda__819,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_afb(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_819
tff(fact_9004_ATP_Olambda__820,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_it(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_820
tff(fact_9005_ATP_Olambda__821,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_mv(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc),aa(C,set(product_prod(A,B)),Uu,Uua)) ) ).

% ATP.lambda_821
tff(fact_9006_ATP_Olambda__822,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_az(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_822
tff(fact_9007_ATP_Olambda__823,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bb(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_823
tff(fact_9008_ATP_Olambda__824,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_sw(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_824
tff(fact_9009_ATP_Olambda__825,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hi(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_825
tff(fact_9010_ATP_Olambda__826,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_jk(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_826
tff(fact_9011_ATP_Olambda__827,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_ahw(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_827
tff(fact_9012_ATP_Olambda__828,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,Uu: fun(C,set(A)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(product_prod(A,B)),aa(C,fun(D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_aih(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu,Uub),aa(D,fun(A,set(B)),aTP_Lamp_aig(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).

% ATP.lambda_828
tff(fact_9013_ATP_Olambda__829,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_qb(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,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,power_power(A,Uua),Uuc)),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_829
tff(fact_9014_ATP_Olambda__830,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A,Uuc: nat] : aa(nat,B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_vz(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uua,Uuc)),aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_830
tff(fact_9015_ATP_Olambda__831,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_pc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu,Uua))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_831
tff(fact_9016_ATP_Olambda__832,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_gt(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,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_832
tff(fact_9017_ATP_Olambda__833,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ur(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> member(B,aa(A,B,Uu,Uuc),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uub),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_833
tff(fact_9018_ATP_Olambda__834,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,A),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_akq(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uuc)
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_834
tff(fact_9019_ATP_Olambda__835,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_ahc(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_835
tff(fact_9020_ATP_Olambda__836,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_gh(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_836
tff(fact_9021_ATP_Olambda__837,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_gc(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_837
tff(fact_9022_ATP_Olambda__838,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_xo(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_838
tff(fact_9023_ATP_Olambda__839,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_adq(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_839
tff(fact_9024_ATP_Olambda__840,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_afp(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_840
tff(fact_9025_ATP_Olambda__841,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_aho(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(B,$o,Uua,Uuc) ) ) ).

% ATP.lambda_841
tff(fact_9026_ATP_Olambda__842,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_aga(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_842
tff(fact_9027_ATP_Olambda__843,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_dx(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_843
tff(fact_9028_ATP_Olambda__844,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_bu(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bt(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_844
tff(fact_9029_ATP_Olambda__845,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_pq(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_845
tff(fact_9030_ATP_Olambda__846,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_pa(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,power_power(real,aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_846
tff(fact_9031_ATP_Olambda__847,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(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,power_power(real,aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_847
tff(fact_9032_ATP_Olambda__848,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_sx(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_848
tff(fact_9033_ATP_Olambda__849,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_acp(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Uu,Uuc))
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc))
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua) ) ) ).

% ATP.lambda_849
tff(fact_9034_ATP_Olambda__850,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_uq(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub)) ) ) ).

% ATP.lambda_850
tff(fact_9035_ATP_Olambda__851,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_tg(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_tf(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_tf(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_tf(A,A)))))) ) ).

% ATP.lambda_851
tff(fact_9036_ATP_Olambda__852,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_td(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_852
tff(fact_9037_ATP_Olambda__853,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_tc(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_853
tff(fact_9038_ATP_Olambda__854,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_mp(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_md(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_854
tff(fact_9039_ATP_Olambda__855,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(list(A),fun(list(A),$o)),Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_aei(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),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) ) )
        | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & aa(A,$o,aa(A,fun(A,$o),Uu,X4),Y4) )
        | ? [X4: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,X4),Y4)
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,Y4),X4)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uua,Xs3),Ys4) ) ) ) ).

% ATP.lambda_855
tff(fact_9040_ATP_Olambda__856,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_aeg(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),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_856
tff(fact_9041_ATP_Olambda__857,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bm($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_857
tff(fact_9042_ATP_Olambda__858,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_ajb(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)),insert(A),Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_858
tff(fact_9043_ATP_Olambda__859,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ma(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ly(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_859
tff(fact_9044_ATP_Olambda__860,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mh(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_md(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_860
tff(fact_9045_ATP_Olambda__861,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_lz(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ly(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_861
tff(fact_9046_ATP_Olambda__862,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_me(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_md(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_862
tff(fact_9047_ATP_Olambda__863,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_tp(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uua)))) ) ) ).

% ATP.lambda_863
tff(fact_9048_ATP_Olambda__864,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_aev(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_864
tff(fact_9049_ATP_Olambda__865,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,fun(list(C),B)),Uub: C,Uuc: list(C)] : aa(list(C),A,aa(C,fun(list(C),A),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_aet(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(list(C),B,aa(C,fun(list(C),B),Uua,Uub),Uuc)) ).

% ATP.lambda_865
tff(fact_9050_ATP_Olambda__866,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_bf(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_866
tff(fact_9051_ATP_Olambda__867,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_867
tff(fact_9052_ATP_Olambda__868,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ja(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_868
tff(fact_9053_ATP_Olambda__869,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_eg(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_869
tff(fact_9054_ATP_Olambda__870,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_cl(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_870
tff(fact_9055_ATP_Olambda__871,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_adr(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_871
tff(fact_9056_ATP_Olambda__872,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(product_prod(B,B),$o),Uuc: A] :
          ( aa(A,$o,aa(fun(product_prod(B,B),$o),fun(A,$o),aa(B,fun(fun(product_prod(B,B),$o),fun(A,$o)),aTP_Lamp_xn(fun(A,B),fun(B,fun(fun(product_prod(B,B),$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(product_prod(B,B),$o,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uuc)),Uua)) ) ) ).

% ATP.lambda_872
tff(fact_9057_ATP_Olambda__873,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_adp(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_873
tff(fact_9058_ATP_Olambda__874,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_vg(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_vf(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_874
tff(fact_9059_ATP_Olambda__875,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_qe(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_qd(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_875
tff(fact_9060_ATP_Olambda__876,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_xp(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_xo(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_876
tff(fact_9061_ATP_Olambda__877,axiom,
    ! [A: $tType,Uu: fun(A,$o),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_ahh(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),takeWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,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_877
tff(fact_9062_ATP_Olambda__878,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_pn(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)))) ) ).

% ATP.lambda_878
tff(fact_9063_ATP_Olambda__879,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_afj(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_879
tff(fact_9064_ATP_Olambda__880,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_alk(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)),insert(B),Uub),bot_bot(set(B)))))) ).

% ATP.lambda_880
tff(fact_9065_ATP_Olambda__881,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_vc(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X4: A] :
          ( ( member(A,X4,Uu)
           => member(B,aa(A,B,Uuc,X4),Uua) )
          & ( ~ member(A,X4,Uu)
           => ( aa(A,B,Uuc,X4) = Uub ) ) ) ) ).

% ATP.lambda_881
tff(fact_9066_ATP_Olambda__882,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_abs(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [Y4: C] :
          ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y4),Uu)
          & 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_882
tff(fact_9067_ATP_Olambda__883,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_aax(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),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)) )
          & member(C,A7,Uu) ) ) ).

% ATP.lambda_883
tff(fact_9068_ATP_Olambda__884,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( aa(C,$o,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_aac(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [X4: A,Y4: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X4),Y4) )
          & aa(A,$o,Uu,X4)
          & aa(B,$o,Uua,Y4) ) ) ).

% ATP.lambda_884
tff(fact_9069_ATP_Olambda__885,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_abv(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_885
tff(fact_9070_ATP_Olambda__886,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_abt(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(product_prod(A,C),fun(set(product_prod(A,C)),set(product_prod(A,C))),insert(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_886
tff(fact_9071_ATP_Olambda__887,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_py(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_px(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_887
tff(fact_9072_ATP_Olambda__888,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_gq(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_gp(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_888
tff(fact_9073_ATP_Olambda__889,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_ov(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_ou(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,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,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_889
tff(fact_9074_ATP_Olambda__890,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_gu(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_gt(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,power_power(A,Uub),Uud)) ) ).

% ATP.lambda_890
tff(fact_9075_ATP_Olambda__891,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hc(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_891
tff(fact_9076_ATP_Olambda__892,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_gy(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uuc),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_892
tff(fact_9077_ATP_Olambda__893,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abg(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_trancl(A,Uub))
          | ( Uuc = Uu ) )
        & ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_trancl(A,Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_893
tff(fact_9078_ATP_Olambda__894,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_gz(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_894
tff(fact_9079_ATP_Olambda__895,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ako(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uuc),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uud),bot_bot(set(A))))))),field2(A,Uu))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))) )
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & ( Uua = Uuc )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))) ) ) ) ) ).

% ATP.lambda_895
tff(fact_9080_ATP_Olambda__896,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_abf(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_rtrancl(A,Uub))
        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_rtrancl(A,Uub)) ) ) ).

% ATP.lambda_896
tff(fact_9081_ATP_Olambda__897,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_gp(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_897
tff(fact_9082_ATP_Olambda__898,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(product_prod(B,B),$o),Uuc: A,Uud: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aa(fun(product_prod(B,B),$o),fun(A,fun(A,$o)),aa(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o))),aTP_Lamp_yz(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( member(A,Uuc,Uu)
           => ( member(A,Uud,Uu)
             => aa(product_prod(B,B),$o,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uuc)),aa(A,B,Uua,Uud))) ) ) ) ) ).

% ATP.lambda_898
tff(fact_9083_ATP_Olambda__899,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A,Uuc: A,Uud: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_xw(fun(product_prod(A,A),$o),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( ( Uub = Uuc )
           => aa(product_prod(A,A),$o,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)) ) ) ) ).

% ATP.lambda_899
tff(fact_9084_ATP_Olambda__900,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: int,Uud: B] : aa(B,A,aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_zp(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uub,Uud)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uuc)),power_int(A,aa(B,A,Uu,Uua),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uuc),one_one(int))))) ) ).

% ATP.lambda_900
tff(fact_9085_ATP_Olambda__901,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_px(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_pv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_901
tff(fact_9086_ATP_Olambda__902,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pl(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = divide_divide(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_902
tff(fact_9087_ATP_Olambda__903,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_qd(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),Uu)
        | ( ( Uub = Uud )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),Uua) ) ) ) ).

% ATP.lambda_903
tff(fact_9088_ATP_Olambda__904,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_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_904
tff(fact_9089_ATP_Olambda__905,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pe(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_905
tff(fact_9090_ATP_Olambda__906,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pj(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_906
tff(fact_9091_ATP_Olambda__907,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pu(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),divide_divide(B,aa(A,B,Uua,Uue),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_907
tff(fact_9092_ATP_Olambda__908,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_vf(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa(A,$o,Uu,Uud)
        & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),aa(A,set(product_prod(B,B)),Uua,Uud)) ) ) ).

% ATP.lambda_908
tff(fact_9093_ATP_Olambda__909,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zv(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uue),Uuf)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuf),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uud))
           => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I4)),X_13)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I4) ) )
        & $ite(
            Uue = Uuf,
            ! [X4: vEBT_VEBT] :
              ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X9) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X4)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),X4)
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_909
tff(fact_9094_ATP_Olambda__910,axiom,
    ! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_nk($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_910
tff(fact_9095_ATP_Olambda__911,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ni($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_911
tff(fact_9096_ATP_Olambda__912,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_aii(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_912
tff(fact_9097_ATP_Olambda__913,axiom,
    ! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_aik(set(D),fun(B,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_913
tff(fact_9098_ATP_Olambda__914,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_ajd(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_914
tff(fact_9099_ATP_Olambda__915,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_ail(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_915
tff(fact_9100_ATP_Olambda__916,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_ahr(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_916
tff(fact_9101_ATP_Olambda__917,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_aiy(set(A),fun(A,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_917
tff(fact_9102_ATP_Olambda__918,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_ahn(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_918
tff(fact_9103_ATP_Olambda__919,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_ms(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_919
tff(fact_9104_ATP_Olambda__920,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_aie(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_920
tff(fact_9105_ATP_Olambda__921,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,fun(B,$o),aTP_Lamp_xs(fun(B,$o),fun(A,fun(B,$o)),Uu),Uua) = Uu ).

% ATP.lambda_921
tff(fact_9106_ATP_Olambda__922,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_aby(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_922
tff(fact_9107_ATP_Olambda__923,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_lr(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_923
tff(fact_9108_ATP_Olambda__924,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_lq(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_924
tff(fact_9109_ATP_Olambda__925,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_pf(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_925
tff(fact_9110_ATP_Olambda__926,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_lp(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_926
tff(fact_9111_ATP_Olambda__927,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ql(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_927
tff(fact_9112_ATP_Olambda__928,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_qj(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_928
tff(fact_9113_ATP_Olambda__929,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_lx(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_929
tff(fact_9114_ATP_Olambda__930,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_mq(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_930
tff(fact_9115_ATP_Olambda__931,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hf(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_931
tff(fact_9116_ATP_Olambda__932,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nx(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_932
tff(fact_9117_ATP_Olambda__933,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_mc(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_933
tff(fact_9118_ATP_Olambda__934,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_akg(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_934
tff(fact_9119_ATP_Olambda__935,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_qk(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_935
tff(fact_9120_ATP_Olambda__936,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_aid(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_936
tff(fact_9121_ATP_Olambda__937,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_afe(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_937
tff(fact_9122_ATP_Olambda__938,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_kj(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_938
tff(fact_9123_ATP_Olambda__939,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_ki(B,fun(A,A)),Uu),Uua) = Uua ).

% ATP.lambda_939
tff(fact_9124_ATP_Olambda__940,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_agy(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_940
tff(fact_9125_ATP_Olambda__941,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_aez(A,fun(list(A),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_941
tff(fact_9126_ATP_Olambda__942,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_aey(A,fun(list(A),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_942
tff(fact_9127_ATP_Olambda__943,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_br(A,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_943
tff(fact_9128_ATP_Olambda__944,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agd(A,fun(A,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_944
tff(fact_9129_ATP_Olambda__945,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_cm(complex,complex),Uu) = Uu ).

% ATP.lambda_945
tff(fact_9130_ATP_Olambda__946,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ca(nat,nat),Uu) = Uu ).

% ATP.lambda_946
tff(fact_9131_ATP_Olambda__947,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_bl(int,int),Uu) = Uu ).

% ATP.lambda_947
tff(fact_9132_ATP_Olambda__948,axiom,
    ! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_afn(B,B),Uu) = Uu ).

% ATP.lambda_948
tff(fact_9133_ATP_Olambda__949,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_tf(A,A),Uu) = Uu ) ).

% ATP.lambda_949
tff(fact_9134_ATP_Olambda__950,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_tj(A,A),Uu) = Uu ) ).

% ATP.lambda_950
tff(fact_9135_ATP_Olambda__951,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_vr(A,A),Uu) = Uu ) ).

% ATP.lambda_951
tff(fact_9136_ATP_Olambda__952,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_afq(A,A),Uu) = Uu ) ).

% ATP.lambda_952
tff(fact_9137_ATP_Olambda__953,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_mr(A,A),Uu) = Uu ) ).

% ATP.lambda_953
tff(fact_9138_ATP_Olambda__954,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ao(A,A),Uu) = Uu ) ).

% ATP.lambda_954
tff(fact_9139_ATP_Olambda__955,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_adm(A,A),Uu) = Uu ).

% ATP.lambda_955
tff(fact_9140_ATP_Olambda__956,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_ahq(A,set(B)),Uu) = top_top(set(B)) ).

% ATP.lambda_956
tff(fact_9141_ATP_Olambda__957,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_yn(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_957
tff(fact_9142_ATP_Olambda__958,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_mw(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_958
tff(fact_9143_ATP_Olambda__959,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_lo(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_959
tff(fact_9144_ATP_Olambda__960,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_ln(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_960
tff(fact_9145_ATP_Olambda__961,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_yo(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_961
tff(fact_9146_ATP_Olambda__962,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_aht(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_962
tff(fact_9147_ATP_Olambda__963,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_en(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_963
tff(fact_9148_ATP_Olambda__964,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_dp(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_964
tff(fact_9149_ATP_Olambda__965,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_bn(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_965
tff(fact_9150_ATP_Olambda__966,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_afo(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_966
tff(fact_9151_ATP_Olambda__967,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_pg(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_967
tff(fact_9152_ATP_Olambda__968,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_an(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_968
tff(fact_9153_ATP_Olambda__969,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_iz(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_969
tff(fact_9154_ATP_Olambda__970,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_acg(B,option(A)),Uu) = none(A) ).

% ATP.lambda_970
tff(fact_9155_ATP_Olambda__971,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_ace(A,option(B)),Uu) = none(B) ).

% ATP.lambda_971
tff(fact_9156_ATP_Olambda__972,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_ais(A,B),Uu) = undefined(B) ).

% ATP.lambda_972
tff(fact_9157_ATP_Olambda__973,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_jy(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_973
tff(fact_9158_ATP_Olambda__974,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_al(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_974
tff(fact_9159_ATP_Olambda__975,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_jz(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_975
tff(fact_9160_ATP_Olambda__976,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_aaa(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_976
tff(fact_9161_ATP_Olambda__977,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_abp(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_977

% Type constructors (746)
tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple592849572758109894attice(A16)
     => counta4013691401010221786attice(fun(A15,A16)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => condit1219197933456340205attice(fun(A15,A16)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( counta3822494911875563373attice(A16)
     => counta3822494911875563373attice(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple592849572758109894attice(A16)
     => comple592849572758109894attice(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bounded_lattice(A16)
     => bounde4967611905675639751up_bot(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bounded_lattice(A16)
     => bounde4346867609351753570nf_top(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => comple6319245703460814977attice(fun(A15,A16)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A15: $tType,A16: $tType] :
      ( boolea8198339166811842893lgebra(A16)
     => boolea8198339166811842893lgebra(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bounded_lattice(A16)
     => bounded_lattice_top(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bounded_lattice(A16)
     => bounded_lattice_bot(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => comple9053668089753744459l_ccpo(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A15: $tType,A16: $tType] :
      ( semilattice_sup(A16)
     => semilattice_sup(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A15: $tType,A16: $tType] :
      ( semilattice_inf(A16)
     => semilattice_inf(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( distrib_lattice(A16)
     => distrib_lattice(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bounded_lattice(A16)
     => bounded_lattice(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order_top(A16)
     => order_top(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order_bot(A16)
     => order_bot(fun(A15,A16)) ) ).

tff(tcon_fun___Countable_Ocountable,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( finite_finite(A15)
        & countable(A16) )
     => countable(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A15: $tType,A16: $tType] :
      ( preorder(A16)
     => preorder(fun(A15,A16)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( finite_finite(A15)
        & finite_finite(A16) )
     => finite_finite(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( lattice(A16)
     => lattice(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order(A16)
     => order(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A15: $tType,A16: $tType] :
      ( top(A16)
     => top(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ord(A16)
     => ord(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bot(A16)
     => bot(fun(A15,A16)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A15: $tType,A16: $tType] :
      ( uminus(A16)
     => uminus(fun(A15,A16)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A15: $tType,A16: $tType] :
      ( minus(A16)
     => minus(fun(A15,A16)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
    condit1219197933456340205attice(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
    topolo8865339358273720382pology(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
    topolo5987344860129210374id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add(int) ).

tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
    topolo4211221413907600880p_mult(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
    topolo6943815403480290642id_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___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___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___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___Parity_Oring__parity,axiom,
    ring_parity(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_Oidom__modulo,axiom,
    idom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Lattices_Olattice_7,axiom,
    lattice(int) ).

tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
    group_add(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
    semiring_gcd(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
    semiring_Gcd(int) ).

tff(tcon_Int_Oint___Rings_Omult__zero,axiom,
    mult_zero(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

tff(tcon_Int_Oint___Orderings_Oorder_8,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_9,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_10,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Groups_Ominus_11,axiom,
    minus(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_12,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_13,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_14,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_15,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_16,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_17,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_18,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_19,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_20,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_21,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_22,axiom,
    strict9044650504122735259up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
    ordere1170586879665033532d_diff(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_23,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_24,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_25,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_26,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_27,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_28,axiom,
    ordere8940638589300402666id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
    canoni5634975068530333245id_add(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_29,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_30,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_31,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_32,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_33,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_34,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_35,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_36,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_37,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_38,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_39,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_40,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_41,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_44,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_45,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_46,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_47,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_48,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_49,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_50,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_51,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_52,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_53,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_54,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_55,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_56,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_57,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_58,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_59,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_60,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_61,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_62,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_63,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_64,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_65,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_66,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_67,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_68,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_69,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_70,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_71,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_72,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_73,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_74,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_75,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_76,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_77,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Countable_Ocountable_78,axiom,
    countable(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_79,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_80,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_81,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_82,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_83,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_84,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_85,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_86,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_87,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_88,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_89,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_90,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_91,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_92,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_93,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_94,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_95,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_96,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_97,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_98,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_99,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_100,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_101,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_102,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_103,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_104,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_105,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_106,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_107,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_108,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_109,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_110,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_111,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_112,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_113,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_114,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_115,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_116,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_117,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_118,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_119,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_120,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_121,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_122,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_123,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_124,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_125,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_126,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_127,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_128,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_129,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_130,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_131,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_132,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_133,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_134,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_135,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_136,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_137,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_138,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_139,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_140,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_141,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_142,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_143,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_144,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_145,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_146,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_147,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_148,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_149,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_150,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_151,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_152,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_153,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_154,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_155,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_156,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_157,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
    field_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_158,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_159,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_160,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_161,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Countable_Ocountable_162,axiom,
    countable(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_163,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_164,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_165,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_166,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_167,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_168,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_169,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_170,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_171,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_172,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_173,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_174,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_175,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_176,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_177,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_178,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_179,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_180,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_181,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_182,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_183,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_184,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_185,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_186,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_187,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_188,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_189,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_190,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_191,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_192,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_193,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_194,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_195,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_196,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_197,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_198,axiom,
    ! [A15: $tType] : counta4013691401010221786attice(set(A15)) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_199,axiom,
    ! [A15: $tType] : condit1219197933456340205attice(set(A15)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_200,axiom,
    ! [A15: $tType] : counta3822494911875563373attice(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_201,axiom,
    ! [A15: $tType] : comple592849572758109894attice(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_202,axiom,
    ! [A15: $tType] : bounde4967611905675639751up_bot(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_203,axiom,
    ! [A15: $tType] : bounde4346867609351753570nf_top(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_204,axiom,
    ! [A15: $tType] : comple6319245703460814977attice(set(A15)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_205,axiom,
    ! [A15: $tType] : boolea8198339166811842893lgebra(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_206,axiom,
    ! [A15: $tType] : bounded_lattice_top(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_207,axiom,
    ! [A15: $tType] : bounded_lattice_bot(set(A15)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_208,axiom,
    ! [A15: $tType] : comple9053668089753744459l_ccpo(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_209,axiom,
    ! [A15: $tType] : semilattice_sup(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_210,axiom,
    ! [A15: $tType] : semilattice_inf(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_211,axiom,
    ! [A15: $tType] : distrib_lattice(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_212,axiom,
    ! [A15: $tType] : bounded_lattice(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_213,axiom,
    ! [A15: $tType] : order_top(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_214,axiom,
    ! [A15: $tType] : order_bot(set(A15)) ).

tff(tcon_Set_Oset___Countable_Ocountable_215,axiom,
    ! [A15: $tType] :
      ( finite_finite(A15)
     => countable(set(A15)) ) ).

tff(tcon_Set_Oset___Orderings_Opreorder_216,axiom,
    ! [A15: $tType] : preorder(set(A15)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_217,axiom,
    ! [A15: $tType] :
      ( finite_finite(A15)
     => finite_finite(set(A15)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_218,axiom,
    ! [A15: $tType] : lattice(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder_219,axiom,
    ! [A15: $tType] : order(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Otop_220,axiom,
    ! [A15: $tType] : top(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oord_221,axiom,
    ! [A15: $tType] : ord(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Obot_222,axiom,
    ! [A15: $tType] : bot(set(A15)) ).

tff(tcon_Set_Oset___Groups_Ouminus_223,axiom,
    ! [A15: $tType] : uminus(set(A15)) ).

tff(tcon_Set_Oset___Groups_Ominus_224,axiom,
    ! [A15: $tType] : minus(set(A15)) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_225,axiom,
    counta4013691401010221786attice($o) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_226,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_227,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_228,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_229,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_230,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_231,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_232,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_233,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_234,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_235,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_236,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_237,axiom,
    bounded_lattice_top($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_238,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_239,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_240,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_241,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_242,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_243,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_244,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_245,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_246,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_247,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Countable_Ocountable_248,axiom,
    countable($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_249,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_250,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_251,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_252,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_253,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_254,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_255,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_256,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_257,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_258,axiom,
    minus($o) ).

tff(tcon_List_Olist___Countable_Ocountable_259,axiom,
    ! [A15: $tType] :
      ( countable(A15)
     => countable(list(A15)) ) ).

tff(tcon_List_Olist___Nat_Osize_260,axiom,
    ! [A15: $tType] : size(list(A15)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_261,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_262,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_263,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_264,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_265,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_266,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_267,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_268,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_269,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_270,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_271,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_272,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_273,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_274,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_275,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_276,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_277,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_278,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_279,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_280,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_281,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_282,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_283,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_284,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_285,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_286,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_287,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_288,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_289,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_290,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_291,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_292,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_293,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_294,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_295,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_296,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_297,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_298,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_299,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_300,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_301,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_302,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_303,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_304,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_305,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_306,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_307,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_308,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_309,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_310,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_311,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_312,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_313,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_314,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_315,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_316,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_317,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_318,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_319,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_320,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_321,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_322,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_323,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_324,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_325,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_326,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_327,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_328,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_329,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_330,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_331,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_332,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_333,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_334,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_335,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_336,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_337,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_338,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_339,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_340,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_341,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_342,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_343,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_344,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_345,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_346,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_347,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_348,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_349,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_350,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_351,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_352,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_353,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_354,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_355,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_356,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_357,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_358,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_359,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_360,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_361,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_362,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_363,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_364,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_365,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_366,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_367,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_368,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_369,axiom,
    dvd(real) ).

tff(tcon_Sum__Type_Osum___Countable_Ocountable_370,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( countable(A15)
        & countable(A16) )
     => countable(sum_sum(A15,A16)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_371,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( finite_finite(A15)
        & finite_finite(A16) )
     => finite_finite(sum_sum(A15,A16)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_372,axiom,
    ! [A15: $tType,A16: $tType] : size(sum_sum(A15,A16)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_373,axiom,
    ! [A15: $tType] : condit1219197933456340205attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_374,axiom,
    ! [A15: $tType] : counta3822494911875563373attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_375,axiom,
    ! [A15: $tType] : bounde4967611905675639751up_bot(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_376,axiom,
    ! [A15: $tType] : bounde4346867609351753570nf_top(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_377,axiom,
    ! [A15: $tType] : comple6319245703460814977attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_378,axiom,
    ! [A15: $tType] : bounded_lattice_top(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_379,axiom,
    ! [A15: $tType] : bounded_lattice_bot(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_380,axiom,
    ! [A15: $tType] : comple9053668089753744459l_ccpo(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_381,axiom,
    ! [A15: $tType] : semilattice_sup(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_382,axiom,
    ! [A15: $tType] : semilattice_inf(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_383,axiom,
    ! [A15: $tType] : distrib_lattice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_384,axiom,
    ! [A15: $tType] : bounded_lattice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_385,axiom,
    ! [A15: $tType] : order_top(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_386,axiom,
    ! [A15: $tType] : order_bot(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_387,axiom,
    ! [A15: $tType] : preorder(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_388,axiom,
    ! [A15: $tType] : lattice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_389,axiom,
    ! [A15: $tType] : order(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_390,axiom,
    ! [A15: $tType] : top(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_391,axiom,
    ! [A15: $tType] : ord(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_392,axiom,
    ! [A15: $tType] : bot(filter(A15)) ).

tff(tcon_Option_Ooption___Countable_Ocountable_393,axiom,
    ! [A15: $tType] :
      ( countable(A15)
     => countable(option(A15)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_394,axiom,
    ! [A15: $tType] :
      ( finite_finite(A15)
     => finite_finite(option(A15)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_395,axiom,
    ! [A15: $tType] : size(option(A15)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_396,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_397,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_398,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_399,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_400,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_401,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_402,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_403,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_404,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_405,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_406,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_407,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_408,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_409,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_410,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_411,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_412,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_413,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_414,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_415,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_416,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_417,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_418,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_419,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_420,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_421,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_422,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_423,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_424,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_425,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_426,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_427,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_428,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_429,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_430,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_431,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_432,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_433,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_434,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_435,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_436,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_437,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_438,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_439,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_440,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_441,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_442,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_443,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_444,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_445,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_446,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_447,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_448,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_449,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_450,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_451,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_452,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_453,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_454,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_455,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_456,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_457,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_458,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_459,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_460,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_461,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_462,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_463,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_464,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_465,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_466,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_467,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_468,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_469,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_470,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_471,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_472,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_473,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_474,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_475,axiom,
    counta4013691401010221786attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_476,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_477,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_478,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_479,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_480,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_481,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_482,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_483,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_484,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_485,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_486,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_487,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_488,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_489,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_490,axiom,
    bounded_lattice_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_491,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_492,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_493,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_494,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_495,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_496,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_497,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_498,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_499,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_500,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_501,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_502,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_503,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_504,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_505,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_506,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_507,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_508,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_509,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_510,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_511,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_512,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_513,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable_Ocountable_514,axiom,
    countable(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_515,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_516,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_517,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_518,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_519,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_520,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_521,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_522,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_523,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_524,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_525,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_526,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_527,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_528,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_529,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_530,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_531,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_532,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_533,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_534,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_535,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_536,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topolo4958980785337419405_space(A15)
        & topolo4958980785337419405_space(A16) )
     => topolo4958980785337419405_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_537,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topological_t2_space(A15)
        & topological_t2_space(A16) )
     => topological_t2_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_538,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topological_t1_space(A15)
        & topological_t1_space(A16) )
     => topological_t1_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Countable_Ocountable_539,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( countable(A15)
        & countable(A16) )
     => countable(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_540,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( finite_finite(A15)
        & finite_finite(A16) )
     => finite_finite(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_541,axiom,
    ! [A15: $tType,A16: $tType] : size(product_prod(A15,A16)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_542,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_543,axiom,
    counta4013691401010221786attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_544,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_545,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_546,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_547,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_548,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_549,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_550,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_551,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_552,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_553,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_554,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_555,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_556,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_557,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_558,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_559,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_560,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_561,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable_Ocountable_562,axiom,
    countable(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_563,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_564,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_565,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_566,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_567,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_568,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_569,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_570,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_571,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_572,axiom,
    minus(product_unit) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_573,axiom,
    size(vEBT_VEBT) ).

% Helper facts (5)
tff(help_fNot_2_1_U,axiom,
    ! [P2: $o] :
      ( (P2)
      | aa($o,$o,fNot,(P2)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P2: $o] :
      ( ~ aa($o,$o,fNot,(P2))
      | ~ (P2) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X10: A,Y9: A] :
      ( ( X10 != Y9 )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y9) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X10: A,Y9: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y9)
      | ( X10 = Y9 ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( aa(A,$o,P2,fChoice(A,P2))
      = ( ? [X9: A] : aa(A,$o,P2,X9) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary)),ya)
    | ( xa = ya ) ) ).

%------------------------------------------------------------------------------