%------------------------------------------------------------------------------
% File     : ITP232_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 00398_025158
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0067_VEBT_InsertCorrectness_00398_025158 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11498 (4003 unt;1565 typ;   0 def)
%            Number of atoms       : 19227 (7959 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 20663 (2154   ~; 367   |;2166   &)
%                                         (2161 <=>;13815  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   14 (   2 avg)
%            Number of FOOLs       :  740 ( 457 fml; 283 var)
%            Number of X terms     :  607 (   0  []; 537 ite;  70 let)
%            Number of types       :   12 (  11 usr)
%            Number of type conns  : 1354 (1149   >; 205   *;   0   +;   0  <<)
%            Number of predicates  :  263 ( 260 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1308 (1308 usr;  49 con; 0-7 aty)
%            Number of variables   : 34901 (31457   !; 982   ?;34901   :)
%                                         (2462  !>;   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 20:55:18.331
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

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

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

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

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

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

tff(ty_t_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 (1546)
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_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_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] : fun(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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_BNF__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_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_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Finite__Set_OFpow,type,
    finite_Fpow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : fun(set(B),nat) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
    finite6289374366891150609ommute: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
    finite4664212375090638736ute_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
    finite673082921795544331dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
    finite4980608107308702382axioms: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : fun(set(A),$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_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_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

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

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

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

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

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

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

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

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

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

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : fun(list(A),set(A)) ).

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

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

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

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Map_Omap__add,type,
    map_add: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

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

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

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,$o) ) ).

tff(sy_c_Nat_Osemiring__1__class_ONats,type,
    semiring_1_Nats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

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

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: 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__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun(real,real) ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun(real,real) ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: 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_Num_Oring__1__class_Oiszero,type,
    ring_1_iszero: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Num_Osqr,type,
    sqr: num > num ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : fun(option(A),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_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] : ( A > fun(B,product_prod(A,B)) ) ).

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

tff(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * product_prod(A,B) ) > product_prod(A,C) ) ).

tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).

tff(sy_c_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Rat_OFract,type,
    fract: ( int * int ) > rat ).

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: rat > $o ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

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_Odependent,type,
    real_V358717886546972837endent: 
      !>[A: $tType] : ( set(A) > $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] : ( ( A * 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: $tType] : ( fun(A,$o) > set(A) ) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : ( ( set(A) * set(A) ) > $o ) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image2: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert2: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

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

tff(sy_c_Set_Opairwise,type,
    pairwise: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

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

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : 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_Oconnected,type,
    topolo1966860045006549960nected: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
    topolo7230453075368039082e_nhds: 
      !>[A: $tType] : ( A > filter(A) ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
    topolo3814608138187158403Cauchy: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
    topolo6773858410816713723filter: 
      !>[A: $tType] : ( filter(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete,type,
    topolo2479028161051973599mplete: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
    topolo6688025880775521714ounded: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
    topolo7806501430040627800ormity: 
      !>[A: $tType] : filter(product_prod(A,A)) ).

tff(sy_c_Topological__Spaces_Ouniformly__continuous__on,type,
    topolo6026614971017936543ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

tff(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun(real,real) ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: real > real ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

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

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_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_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) * set(A) ) > $o ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Owf,type,
    wf: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_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_i____,type,
    i: 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 ).

% Relevant facts (9099)
tff(fact_0_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Ta)
     => ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_1) ) ).

% not_min_Null_member
tff(fact_1_False,axiom,
    ~ vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(mi,na))) ).

% False
tff(fact_2__092_060open_062_Iif_AminNull_A_ItreeList_A_B_Ahigh_Ami_An_J_Athen_Avebt__insert_Asummary_A_Ihigh_Ami_An_J_Aelse_Asummary_J_A_061_Asummary_092_060close_062,axiom,
    $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) = summary ).

% \<open>(if minNull (treeList ! high mi n) then vebt_insert summary (high mi n) else summary) = summary\<close>
tff(fact_3__C8_C,axiom,
    na = m ).

% "8"
tff(fact_4__C12_C,axiom,
    vEBT_invar_vebt($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),na) ).

% "12"
tff(fact_5__C5_C,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) ) ).

% "5"
tff(fact_6_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_7__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_8__C0_C,axiom,
    ! [X: vEBT_VEBT] :
      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => vEBT_invar_vebt(X,na) ) ).

% "0"
tff(fact_9_mimaxrel,axiom,
    ( ( xa != mi )
    & ( xa != ma ) ) ).

% mimaxrel
tff(fact_10__C11_C,axiom,
    ! [X: vEBT_VEBT] :
      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),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)))))
     => vEBT_invar_vebt(X,na) ) ).

% "11"
tff(fact_11_bit__split__inv,axiom,
    ! [Xb: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xb,D2),vEBT_VEBT_low(Xb,D2),D2) = Xb ).

% bit_split_inv
tff(fact_12_abcdef,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),xa),mi) ).

% abcdef
tff(fact_13__C3_C,axiom,
    deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) ).

% "3"
tff(fact_14_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_15__C1_C,axiom,
    vEBT_invar_vebt(summary,m) ).

% "1"
tff(fact_16_nth__list__update__neq,axiom,
    ! [A: $tType,Ia: nat,J: nat,Xs: list(A),Xb: A] :
      ( ( Ia != J )
     => ( aa(nat,A,nth(A,list_update(A,Xs,Ia,Xb)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_17_list__update__id,axiom,
    ! [A: $tType,Xs: list(A),Ia: nat] : list_update(A,Xs,Ia,aa(nat,A,nth(A,Xs),Ia)) = Xs ).

% list_update_id
tff(fact_18_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_19_nat__add__left__cancel__le,axiom,
    ! [K: 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),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_20_nat__add__left__cancel__less,axiom,
    ! [K: 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),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_21_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_22_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_23_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_24_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_25_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_26_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_27_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_leaf
tff(fact_28_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_29_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_30_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_31_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_32_list__update__overwrite,axiom,
    ! [A: $tType,Xs: list(A),Ia: nat,Xb: A,Y: A] : list_update(A,list_update(A,Xs,Ia,Xb),Ia,Y) = list_update(A,Xs,Ia,Y) ).

% list_update_overwrite
tff(fact_33_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_34_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B3)
    <=> ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => member(A,X2,B3) ) ) ).

% subset_code(1)
tff(fact_35_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),Xb: A,Ia: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
     => ( member(A,Xb,A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,Ia,Xb))),A3) ) ) ).

% set_update_subsetI
tff(fact_36_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X3: A] :
              ( ! [Y2: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X3))
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X3) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_37_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X3: A] :
              ( ! [Y2: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X3))
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X3) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_38_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_39_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_40_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.left_commute
tff(fact_41_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_42_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_43_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_44_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_45_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      ( member(A,A2,collect(A,P))
    <=> aa(A,$o,P,A2) ) ).

% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,$o),A3)) = A3 ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
        <=> aa(A,$o,Q,X3) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_48_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X3: A] : aa(A,B,F2,X3) = aa(A,B,G,X3)
     => ( F2 = G ) ) ).

% ext
tff(fact_49_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B3: A,K: A,B2: A,A2: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_50_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_51_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( ( Ia = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_52_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_53_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V: fun(A,nat),Xb: A] :
      ( ! [X3: A] :
          ( ~ aa(A,$o,P,X3)
         => ? [Y2: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y2)),aa(A,nat,V,X3))
              & ~ aa(A,$o,P,Y2) ) )
     => aa(A,$o,P,Xb) ) ).

% infinite_descent_measure
tff(fact_54_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_55_infinite__descent,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
              & ~ aa(nat,$o,P,M) ) )
     => aa(nat,$o,P,Nb) ) ).

% infinite_descent
tff(fact_56_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
             => aa(nat,$o,P,M) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% nat_less_induct
tff(fact_57_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_irrefl_nat
tff(fact_58_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_59_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_60_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_not_refl
tff(fact_61_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_62_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => ? [X3: nat] :
            ( aa(nat,$o,P,X3)
            & ! [Y2: nat] :
                ( aa(nat,$o,P,Y2)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_63_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_64_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_65_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_66_le__trans,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),K) ) ) ).

% le_trans
tff(fact_67_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Nb) ).

% le_refl
tff(fact_68_list__update__swap,axiom,
    ! [A: $tType,Ia: nat,I: nat,Xs: list(A),Xb: A,X4: A] :
      ( ( Ia != I )
     => ( list_update(A,list_update(A,Xs,Ia,Xb),I,X4) = list_update(A,list_update(A,Xs,I,X4),Ia,Xb) ) ) ).

% list_update_swap
tff(fact_69_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_70_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_71_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)
        <=> ? [C3: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ) ).

% le_iff_add
tff(fact_72_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_right_mono
tff(fact_73_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)
         => ~ ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ).

% less_eqE
tff(fact_74_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_left_mono
tff(fact_75_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_mono
tff(fact_76_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_77_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( ( Ia = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_78_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_79_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_80_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_81_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_strict_right_mono
tff(fact_82_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_strict_left_mono
tff(fact_83_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_strict_mono
tff(fact_84_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_85_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( ( Ia = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_86_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_87_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),Ia: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,I2)),aa(nat,nat,F2,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F2,Ia)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_88_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_89_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_90_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_91_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_92_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_93_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_94_trans__less__add2,axiom,
    ! [Ia: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),J)) ) ).

% trans_less_add2
tff(fact_95_trans__less__add1,axiom,
    ! [Ia: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Mb)) ) ).

% trans_less_add1
tff(fact_96_add__less__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_less_mono1
tff(fact_97_not__add__less2,axiom,
    ! [J: nat,Ia: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Ia)),Ia) ).

% not_add_less2
tff(fact_98_not__add__less1,axiom,
    ! [Ia: 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),Ia),J)),Ia) ).

% not_add_less1
tff(fact_99_add__less__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_100_add__lessD1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),K) ) ).

% add_lessD1
tff(fact_101_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
    <=> ? [K2: nat] : Nb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2) ) ).

% nat_le_iff_add
tff(fact_102_trans__le__add2,axiom,
    ! [Ia: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),J)) ) ).

% trans_le_add2
tff(fact_103_trans__le__add1,axiom,
    ! [Ia: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Mb)) ) ).

% trans_le_add1
tff(fact_104_add__le__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_105_add__le__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_106_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
     => ? [N: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) ) ).

% le_Suc_ex
tff(fact_107_add__leD2,axiom,
    ! [Mb: nat,K: 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),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% add_leD2
tff(fact_108_add__leD1,axiom,
    ! [Mb: nat,K: 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),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% add_leD1
tff(fact_109_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_110_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_111_add__leE,axiom,
    ! [Mb: nat,K: 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),K)),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),K),Nb) ) ) ).

% add_leE
tff(fact_112_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_113_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_114_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_115_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Ia: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_116_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),Mb: nat,K: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M2)),aa(nat,nat,F2,N)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,Mb)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K))) ) ).

% mono_nat_linear_lb
tff(fact_117_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_118_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X3: A] :
            ( aa(A,$o,P,X3)
           => ? [Y2: A] :
                ( aa(A,$o,P,Y2)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y2)),aa(A,nat,F2,X3)) ) )
       => ? [Y3: A] :
            ( aa(A,$o,P,Y3)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_119_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_120_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
     => vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq2
tff(fact_121_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(Ta,D2)
     => vEBT_VEBT_valid(Ta,D2) ) ).

% valid_eq1
tff(fact_122_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
    <=> vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq
tff(fact_123_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_124_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_125_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Nb: nat] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% inthall
tff(fact_126_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_127_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y3)),B2) )
       => ? [X3: A] :
            ( aa(A,$o,P,X3)
            & ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y2)),aa(A,nat,F2,X3)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_128_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun(nat,$o),Mb: nat] :
      ( ! [K3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K3)
         => aa(nat,$o,P,K3) )
     => ( ! [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),Nb)
           => ( ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),I3)
                 => aa(nat,$o,P,I3) )
             => aa(nat,$o,P,K3) ) )
       => aa(nat,$o,P,Mb) ) ) ).

% nat_descend_induct
tff(fact_129_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_130_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_140_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_141_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_142_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_143_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_144_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_145_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_146_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% le0
tff(fact_147_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_148_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_149_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),Ia: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,Ia,Xb)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),Ia: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Ia)
     => ( list_update(A,Xs,Ia,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_164_nth__list__update__eq,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,Ia,Xb)),Ia) = Xb ) ) ).

% nth_list_update_eq
tff(fact_165_set__swap,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$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,Ia,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),Ia))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_166_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_167_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_168_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: A] :
          ( ( zero_zero(A) = Xb )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_169_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xb) != aa(A,nat,size_size(A),Y) )
         => ( Xb != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_170_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_171_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$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),$o,P,Ys2) )
         => aa(list(A),$o,P,Xs2) )
     => aa(list(A),$o,P,Xs) ) ).

% length_induct
tff(fact_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% not_less0
tff(fact_184_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_185_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_186_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P,N)
             => ? [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
                  & ~ aa(nat,$o,P,M) ) ) )
       => aa(nat,$o,P,Nb) ) ) ).

% infinite_descent0
tff(fact_187_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,$o),Xb: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V,X3) = zero_zero(nat) )
         => aa(A,$o,P,X3) )
     => ( ! [X3: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X3))
           => ( ~ aa(A,$o,P,X3)
             => ? [Y2: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y2)),aa(A,nat,V,X3))
                  & ~ aa(A,$o,P,Y2) ) ) )
       => aa(A,$o,P,Xb) ) ) ).

% infinite_descent0_measure
tff(fact_188_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_189_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_190_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_191_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_192_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_193_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_194_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_195_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
         => ? [X_13: A] : aa(A,$o,aa(nat,fun(A,$o),P,I4),X_13) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I4),aa(nat,A,nth(A,Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_196_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_197_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_198_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing
tff(fact_199_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$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),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_200_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$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),C2)) ) ) ) ).

% add_increasing2
tff(fact_201_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_202_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_203_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_204_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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),Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_205_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_206_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_207_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)
         => ~ ! [C4: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) )
               => ( C4 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_208_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% pos_add_strict
tff(fact_209_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),Nb)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K3)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,K3) ) ) ) ).

% ex_least_nat_le
tff(fact_210_less__imp__add__positive,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ? [K3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K3)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_211_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_212_list__ball__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$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,$o,P,X3) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_213_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_214_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xb: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_215_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_216_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_217_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
             => ~ aa(A,$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,$o),ord_less_eq(A),Z),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
                   => ~ aa(A,$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),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_218_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ) ).

% order_class.order_eq_iff
tff(fact_219_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_220_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_221_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
           => ( Xb = Y ) ) ) ) ).

% order_antisym
tff(fact_222_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% order.trans
tff(fact_223_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) ) ) ) ).

% order_trans
tff(fact_224_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$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),P,A4),B4) )
         => ( ! [A4: A,B4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B4),A4)
               => aa(A,$o,aa(A,fun(A,$o),P,A4),B4) )
           => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_225_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_226_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_227_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% dual_order.trans
tff(fact_228_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_229_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funD
tff(fact_230_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funE
tff(fact_231_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).

% le_funI
tff(fact_232_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
        <=> ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) ) ) ).

% le_fun_def
tff(fact_233_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_234_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X3: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_subst1
tff(fact_235_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X3: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_subst2
tff(fact_236_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% order_eq_refl
tff(fact_237_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_linear
tff(fact_238_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X3: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_239_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X3: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_240_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_le_cases
tff(fact_241_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_242_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_243_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb) ) ).

% lt_ex
tff(fact_244_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xb: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X_1) ) ).

% gt_ex
tff(fact_245_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Y) ) ) ) ).

% dense
tff(fact_246_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Xb != Y ) ) ) ).

% less_imp_neq
tff(fact_247_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_248_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_249_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_250_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [X3: A] :
              ( ! [Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X3)
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X3) )
         => aa(A,$o,P,A2) ) ) ).

% less_induct
tff(fact_251_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv3
tff(fact_252_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ( Xb != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_cases
tff(fact_253_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_254_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_255_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
        <=> ? [N2: A] :
              ( aa(A,$o,P,N2)
              & ! [M3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M3),N2)
                 => ~ aa(A,$o,P,M3) ) ) ) ) ).

% exists_least_iff
tff(fact_256_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$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),P,A4),B4) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P,A4),A4)
           => ( ! [A4: A,B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P,A4),B4) )
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_257_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans
tff(fact_258_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
            | ( Xb = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_259_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_260_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_261_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_262_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neqE
tff(fact_263_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_asym
tff(fact_264_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neq_iff
tff(fact_265_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_266_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_less_trans
tff(fact_267_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X3: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_268_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X3: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_269_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_270_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X3: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_271_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X3: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_272_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_not_sym
tff(fact_273_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_274_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          | ( Xb = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_less_linear
tff(fact_275_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Xb != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_276_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Y != Xb ) ) ) ).

% order_less_imp_not_eq2
tff(fact_277_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_imp_not_less
tff(fact_278_nth__list__update,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A),Xb: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,Ia,Xb)),J) = $ite(Ia = J,Xb,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_279_list__update__same__conv,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,Ia,Xb) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),Ia) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_280_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_281_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_282_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_283_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_284_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_285_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_286_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,Mb: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Mb,X3)),aa(A,nat,Mb,Y2)) ) ) ) ).

% ex_has_least_nat
tff(fact_287_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% leD
tff(fact_288_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% leI
tff(fact_289_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_290_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv1
tff(fact_291_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv2
tff(fact_292_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_ge
tff(fact_293_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_le
tff(fact_294_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ) ).

% less_le_not_le
tff(fact_295_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% not_le_imp_less
tff(fact_296_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_297_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_298_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans1
tff(fact_299_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans2
tff(fact_300_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_301_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$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,$o),ord_less_eq(A),Y),W) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_302_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_303_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_304_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_305_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_306_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_307_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_308_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_309_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_310_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_less
tff(fact_311_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & ( Xb != Y ) ) ) ) ).

% order_less_le
tff(fact_312_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_not_le
tff(fact_313_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_not_less
tff(fact_314_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% order_less_imp_le
tff(fact_315_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_316_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_317_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_le_less_trans
tff(fact_318_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_less_le_trans
tff(fact_319_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X3: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_320_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X3: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_321_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X3: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_322_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X3: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_323_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_le_less_linear
tff(fact_324_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_325_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_326_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_epsilon
tff(fact_327_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_328_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_329_subsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X3: A] :
          ( member(A,X3,A3)
         => member(A,X3,B3) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% subsetI
tff(fact_330_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( ( A3 != B3 )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3) ) ) ).

% psubsetI
tff(fact_331_subset__antisym,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( A3 = B3 ) ) ) ).

% subset_antisym
tff(fact_332_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_333_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B4)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B4),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A4),zero_zero(nat))
       => ( ! [A4: nat,B4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B4)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B4)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_334_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_335_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_336_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_337_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_338_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( bot_bot(set(A)) = collect(A,P) )
    <=> ! [X2: A] : ~ aa(A,$o,P,X2) ) ).

% empty_Collect_eq
tff(fact_339_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( collect(A,P) = bot_bot(set(A)) )
    <=> ! [X2: A] : ~ aa(A,$o,P,X2) ) ).

% Collect_empty_eq
tff(fact_340_all__not__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X2: A] : ~ member(A,X2,A3)
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_341_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ member(A,C2,bot_bot(set(A))) ).

% empty_iff
tff(fact_342_empty__subsetI,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A3) ).

% empty_subsetI
tff(fact_343_subset__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A)))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_344_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_345_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F2),G)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F2) ) ) ) ).

% less_fun_def
tff(fact_346_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_347_ex__in__conv,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ? [X2: A] : member(A,X2,A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_348_equals0I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [Y3: A] : ~ member(A,Y3,A3)
     => ( A3 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_349_equals0D,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ( A3 = bot_bot(set(A)) )
     => ~ member(A,A2,A3) ) ).

% equals0D
tff(fact_350_emptyE,axiom,
    ! [A: $tType,A2: A] : ~ member(A,A2,bot_bot(set(A))) ).

% emptyE
tff(fact_351_not__psubset__empty,axiom,
    ! [A: $tType,A3: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_352_psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B3),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C5) ) ) ).

% psubset_trans
tff(fact_353_psubsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( member(A,C2,A3)
       => member(A,C2,B3) ) ) ).

% psubsetD
tff(fact_354_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_355_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_356_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_357_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_358_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_359_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_360_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_361_finite__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A3 ) ).

% finite_list
tff(fact_362_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_363_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_364_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_365_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X_1) ) ).

% linordered_field_no_ub
tff(fact_366_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X) ) ).

% linordered_field_no_lb
tff(fact_367_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_368_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% is_num_normalize(1)
tff(fact_369_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
        | ( A3 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_370_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B3),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C5) ) ) ).

% subset_psubset_trans
tff(fact_371_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ) ).

% subset_not_subset_eq
tff(fact_372_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),C5) ) ) ).

% psubset_subset_trans
tff(fact_373_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% psubset_imp_subset
tff(fact_374_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,P)),collect(A,Q))
    <=> ! [X2: A] :
          ( aa(A,$o,P,X2)
         => aa(A,$o,Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_375_set__eq__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ) ).

% set_eq_subset
tff(fact_376_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5) ) ) ).

% subset_trans
tff(fact_377_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
         => aa(A,$o,Q,X3) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,P)),collect(A,Q)) ) ).

% Collect_mono
tff(fact_378_subset__refl,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),A3) ).

% subset_refl
tff(fact_379_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ! [T2: A] :
          ( member(A,T2,A3)
         => member(A,T2,B3) ) ) ).

% subset_iff
tff(fact_380_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
        & ( A3 != B3 ) ) ) ).

% psubset_eq
tff(fact_381_equalityD2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ).

% equalityD2
tff(fact_382_equalityD1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% equalityD1
tff(fact_383_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ! [X2: A] :
          ( member(A,X2,A3)
         => member(A,X2,B3) ) ) ).

% subset_eq
tff(fact_384_equalityE,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ) ).

% equalityE
tff(fact_385_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ) ).

% psubsetE
tff(fact_386_subsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( member(A,C2,A3)
       => member(A,C2,B3) ) ) ).

% subsetD
tff(fact_387_in__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),Xb: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( member(A,Xb,A3)
       => member(A,Xb,B3) ) ) ).

% in_mono
tff(fact_388_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_389_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B5: A,A5: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B5),A5)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),B5) ) ) ).

% verit_comp_simplify1(3)
tff(fact_390_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,A3)
                & ! [Xa: A] :
                    ( member(A,Xa,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa)
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_391_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,A3)
                & ! [Xa: A] :
                    ( member(A,Xa,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X3)
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_392_finite__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A3: set(A)] : aa(set(A),$o,finite_finite2(A),A3) ) ).

% finite_code
tff(fact_393_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N3)
    <=> ? [M3: nat] :
        ! [X2: nat] :
          ( member(nat,X2,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),M3) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_394_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ! [M3: nat] :
        ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N2)
          & member(nat,N2,S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_395_finite__nat__set__iff__bounded,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N3)
    <=> ? [M3: nat] :
        ! [X2: nat] :
          ( member(nat,X2,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_396_infinite__nat__iff__unbounded,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ! [M3: nat] :
        ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
          & member(nat,N2,S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_397_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_398_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set(nat)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M2)
         => ? [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N4)
              & member(nat,N4,S2) ) )
     => ~ aa(set(nat),$o,finite_finite2(nat),S2) ) ).

% unbounded_k_infinite
tff(fact_399_finite__psubset__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [A6: set(A)] :
            ( aa(set(A),$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),$o,P,B6) )
             => aa(set(A),$o,P,A6) ) )
       => aa(set(A),$o,P,A3) ) ) ).

% finite_psubset_induct
tff(fact_400_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$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,F2,X)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S2))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_401_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_402_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_403_finite__maxlen,axiom,
    ! [A: $tType,M4: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M4)
     => ? [N: 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)),N) ) ) ).

% finite_maxlen
tff(fact_404_finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A3: set(A)] : aa(set(A),$o,finite_finite2(A),A3) ) ).

% finite
tff(fact_405_finite__set__choice,axiom,
    ! [B: $tType,A: $tType,A3: set(A),P: fun(A,fun(B,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [X3: A] :
            ( member(A,X3,A3)
           => ? [X_12: B] : aa(B,$o,aa(A,fun(B,$o),P,X3),X_12) )
       => ? [F3: fun(A,B)] :
          ! [X: A] :
            ( member(A,X,A3)
           => aa(B,$o,aa(A,fun(B,$o),P,X),aa(A,B,F3,X)) ) ) ) ).

% finite_set_choice
tff(fact_406_bounded__Max__nat,axiom,
    ! [P: fun(nat,$o),Xb: nat,M4: nat] :
      ( aa(nat,$o,P,Xb)
     => ( ! [X3: nat] :
            ( aa(nat,$o,P,X3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M4) )
       => ~ ! [M2: nat] :
              ( aa(nat,$o,P,M2)
             => ~ ! [X: nat] :
                    ( aa(nat,$o,P,X)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M2) ) ) ) ) ).

% bounded_Max_nat
tff(fact_407_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => member(A,lattic7623131987881927897min_on(A,B,F2,S2),S2) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_408_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => ? [X3: A] :
                ( member(A,X3,A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3)
                & ! [Xa: A] :
                    ( member(A,Xa,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa)
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_409_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => ? [X3: A] :
                ( member(A,X3,A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A2)
                & ! [Xa: A] :
                    ( member(A,Xa,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X3)
                     => ( X3 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_410_finite_OemptyI,axiom,
    ! [A: $tType] : aa(set(A),$o,finite_finite2(A),bot_bot(set(A))) ).

% finite.emptyI
tff(fact_411_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_412_finite__transitivity__chain,axiom,
    ! [A: $tType,A3: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [X3: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,X3),X3)
       => ( ! [X3: A,Y3: A,Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R,X3),Y3)
             => ( aa(A,$o,aa(A,fun(A,$o),R,Y3),Z2)
               => aa(A,$o,aa(A,fun(A,$o),R,X3),Z2) ) )
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => ? [Y2: A] :
                    ( member(A,Y2,A3)
                    & aa(A,$o,aa(A,fun(A,$o),R,X3),Y2) ) )
           => ( A3 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_413_finite__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% finite_subset
tff(fact_414_infinite__super,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
     => ( ~ aa(set(A),$o,finite_finite2(A),S2)
       => ~ aa(set(A),$o,finite_finite2(A),T3) ) ) ).

% infinite_super
tff(fact_415_rev__finite__subset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
       => aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% rev_finite_subset
tff(fact_416_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),Y: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( member(A,Y,S2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S2))),aa(A,B,F2,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_417_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X3: A] : ~ member(A,X3,A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_418_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D22)
           => ? [E: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_419_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_420_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_421_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_422_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_423_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,P,A2)
           => ( ~ aa(A,$o,P,B2)
             => ? [C4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C4),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),C4) )
                     => aa(A,$o,P,X) )
                  & ! [D3: A] :
                      ( ! [X3: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D3) )
                         => aa(A,$o,P,X3) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C4) ) ) ) ) ) ) ).

% complete_interval
tff(fact_424__C10_C,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),product_Pair(nat,nat,mi),ma)),deg,treeList,summary),xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,xa),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),mi),ma))),deg,list_update(vEBT_VEBT,treeList,vEBT_VEBT_high(mi,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)) ).

% "10"
tff(fact_425__C7_C,axiom,
    ( ( mi != ma )
   => ! [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))),m))
       => ( ( ( vEBT_VEBT_high(ma,na) = I3 )
           => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3)),vEBT_VEBT_low(ma,na)) )
          & ! [Y2: nat] :
              ( ( ( vEBT_VEBT_high(Y2,na) = I3 )
                & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3)),vEBT_VEBT_low(Y2,na)) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),mi),Y2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),ma) ) ) ) ) ) ).

% "7"
tff(fact_426_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_427_intind,axiom,
    ! [A: $tType,Ia: nat,Nb: nat,P: fun(A,$o),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),Nb)
     => ( aa(A,$o,P,Xb)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Nb,Xb)),Ia)) ) ) ).

% intind
tff(fact_428_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_429__C2_C,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) ).

% "2"
tff(fact_430__092_060open_062i_A_060_A2_A_094_Am_092_060close_062,axiom,
    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)) ).

% \<open>i < 2 ^ m\<close>
tff(fact_431_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Mb: num,Nb: num] :
          ( ( aa(num,A,numeral_numeral(A),Mb) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( Mb = Nb ) ) ) ).

% numeral_eq_iff
tff(fact_432_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( bit0(X22) = bit0(Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_433__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_434__C4_Oprems_C,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"
tff(fact_435__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))
           => vEBT_invar_vebt(vEBT_vebt_insert(X,Xa),na) ) ) ) ).

% "4.IH"(1)
tff(fact_436_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),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_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) )
          & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_437__C4_C,axiom,
    ! [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))),m))
     => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I3)),X_13)
      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(summary),I3) ) ) ).

% "4"
tff(fact_438_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_439_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_440_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_441_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_442_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V2: num,W2: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W2)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),W2))),Z) ) ).

% add_numeral_left
tff(fact_443_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_444_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),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),product_Pair(nat,nat,Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_445_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: 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),Y),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,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_446_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_447_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_448_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_449_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: 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),Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
         => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_450_replicate__eq__replicate,axiom,
    ! [A: $tType,Mb: nat,Xb: A,Nb: nat,Y: A] :
      ( ( replicate(A,Mb,Xb) = replicate(A,Nb,Y) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero(nat) )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_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),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_459__C4_OIH_C_I2_J,axiom,
    ! [Xb: 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))
     => vEBT_invar_vebt(vEBT_vebt_insert(summary,Xb),m) ) ).

% "4.IH"(2)
tff(fact_460_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),V2),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_461_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_462_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_463__C6_C,axiom,
    ( 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)) ) ).

% "6"
tff(fact_464_myIHs,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( member(vEBT_VEBT,Xb,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => ( vEBT_invar_vebt(Xb,na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),na))
         => vEBT_invar_vebt(vEBT_vebt_insert(Xb,Xaa),na) ) ) ) ).

% myIHs
tff(fact_465_in__set__replicate,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Y: A] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),replicate(A,Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_466_Bex__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ? [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
          & aa(A,$o,P,X2) )
    <=> ( aa(A,$o,P,A2)
        & ( Nb != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_467_Ball__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
         => aa(A,$o,P,X2) )
    <=> ( aa(A,$o,P,A2)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_468_nth__replicate,axiom,
    ! [A: $tType,Ia: nat,Nb: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),Nb)
     => ( aa(nat,A,nth(A,replicate(A,Nb,Xb)),Ia) = Xb ) ) ).

% nth_replicate
tff(fact_469_highlowprop,axiom,
    ( 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))),m))
    & 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)) ) ).

% highlowprop
tff(fact_470_add__One__commute,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb) = aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2) ).

% add_One_commute
tff(fact_471_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != bit0(X22) ).

% verit_eq_simplify(10)
tff(fact_472_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Xb),one2)
    <=> ( Xb = one2 ) ) ).

% le_num_One_iff
tff(fact_473_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_474_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_475_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y) = Xb ) ) ) ).

% max_absorb1
tff(fact_476_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_477_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: 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),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ).

% max_add_distrib_right
tff(fact_478_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),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),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_479_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_480_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_481_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_482_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_483_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_484_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_485_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_486_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_487_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y3: A] :
            ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate(A,Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_488_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_489_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_490_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
             => ( aa(A,$o,P,X3)
              <=> aa(A,$o,P2,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q3,X3) ) )
           => ? [Z2: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
               => ( ( aa(A,$o,P,X)
                    & aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P2,X)
                    & aa(A,$o,Q3,X) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_491_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
             => ( aa(A,$o,P,X3)
              <=> aa(A,$o,P2,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q3,X3) ) )
           => ? [Z2: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
               => ( ( aa(A,$o,P,X)
                    | aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P2,X)
                    | aa(A,$o,Q3,X) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_492_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_493_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_494_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_495_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_496_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_497_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
             => ( aa(A,$o,P,X3)
              <=> aa(A,$o,P2,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q3,X3) ) )
           => ? [Z2: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
               => ( ( aa(A,$o,P,X)
                    & aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P2,X)
                    & aa(A,$o,Q3,X) ) ) ) ) ) ) ).

% minf(1)
tff(fact_498_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q3: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
             => ( aa(A,$o,P,X3)
              <=> aa(A,$o,P2,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q3,X3) ) )
           => ? [Z2: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
               => ( ( aa(A,$o,P,X)
                    | aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P2,X)
                    | aa(A,$o,Q3,X) ) ) ) ) ) ) ).

% minf(2)
tff(fact_499_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_500_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_501_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_502_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_503_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_504_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_505_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) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,Nb) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(X3,Nb)) )
                                   => ( aa(nat,$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),product_Pair(nat,nat,Mia),Maa)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_506_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: 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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_507_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_508_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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)),Y)
           => ( ( aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( Xb = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_509_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_510_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),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),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_511_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),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),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_512_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_513_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_514_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_515_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa(nat,$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_516_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_517_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_518__C9_C,axiom,
    divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),bit0(one2))) = na ).

% "9"
tff(fact_519_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% div_0
tff(fact_520_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_521_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_522_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( divide_divide(A,C2,A2) = divide_divide(A,C2,B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_523_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_524_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_525_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_526_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_527_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_528_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ).

% add_divide_distrib
tff(fact_529_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_530_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_right_mono
tff(fact_531_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_532_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_533_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_534_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_535_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_536_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),divide_divide(A,A2,C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_537_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_neg_neg
tff(fact_538_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_539_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_540_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_pos_pos
tff(fact_541_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_542_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_543_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_544_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_545_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_546_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : divide_divide(A,A2,aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_547_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$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,Y,W2)) ) ) ) ) ) ).

% frac_le
tff(fact_548_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => ( aa(A,$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,Y,W2)) ) ) ) ) ) ).

% frac_less
tff(fact_549_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$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,Y,W2)) ) ) ) ) ) ).

% frac_less2
tff(fact_550_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_551_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_552_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_553_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_554_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_555_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_556_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_557_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_558_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$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),Y),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_559_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_560_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_561_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_562_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_563_nat__power__less__imp__less,axiom,
    ! [Ia: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ia)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,Ia),Mb)),aa(nat,nat,power_power(nat,Ia),Nb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_564_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_565_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_566_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_567_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_568_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_569_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_570_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_571_self__le__ge2__pow,axiom,
    ! [K: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,power_power(nat,K),Mb)) ) ).

% self_le_ge2_pow
tff(fact_572_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_573_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_574_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),product_Pair(nat,nat,Mia),Maa)),zero_zero(nat),Va,Vb),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_575_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_576_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$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)),Y)
             => ( Xb = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_577_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_578_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_579_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_580_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_581_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : aa(A,$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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_power2_ge_zero
tff(fact_582_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_583_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : ~ aa(A,$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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_584_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( ( Xb != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_585_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_586_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_587_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),product_Pair(nat,nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_588_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_589_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,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa(nat,$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_590_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_591_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_592_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_593_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_594_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_595_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_596_bit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H,L,D2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),D2))),L) ).

% bit_concat_def
tff(fact_597_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_598_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_599_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_600_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_601_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_602_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(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),times_times(num),Mb),Nb)) ) ).

% numeral_times_numeral
tff(fact_603_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V2: num,W2: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_604_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_605_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_606_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_607_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_608_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: 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),Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),Xb),Nb) = Y ) ) ).

% high_inv
tff(fact_609_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_610_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: 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),Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),Xb),Nb) = Xb ) ) ).

% low_inv
tff(fact_611_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C2)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C2) ) ).

% times_divide_eq_left
tff(fact_612_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,divide_divide(A,A2,B2),C2) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_613_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,A2,divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2) ) ).

% divide_divide_eq_right
tff(fact_614_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% times_divide_eq_right
tff(fact_615_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,A2,one_one(A)) = A2 ) ).

% div_by_1
tff(fact_616_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_617_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_618_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_619_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_620_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_621_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_622_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_623_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_624_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_625_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_626_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_627_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_628_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_629_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),divide_divide(A,A2,B2)) ) ).

% div_mult_mult1_if
tff(fact_630_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult2
tff(fact_631_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult1
tff(fact_632_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),divide_divide(A,A2,B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_633_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_634_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_635_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_636_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_637_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_638_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_639_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).

% distrib_right_numeral
tff(fact_640_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).

% distrib_left_numeral
tff(fact_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_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_649_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_650_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(num,A,numeral_numeral(A),Nb) = one_one(A) )
        <=> ( Nb = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_651_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( one2 = Nb ) ) ) ).

% one_eq_numeral_iff
tff(fact_652_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_653_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_654_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_655_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_656_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_657_nat__mult__less__cancel__disj,axiom,
    ! [K: 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),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_658_mult__less__cancel2,axiom,
    ! [Mb: nat,K: 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),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_659_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_660_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_661_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = $ite(K = zero_zero(nat),zero_zero(nat),divide_divide(nat,Mb,Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_662_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_663_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_664_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_665_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_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self4
tff(fact_679_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self3
tff(fact_680_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self2
tff(fact_681_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self1
tff(fact_682_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,B2),Xb)),aa(nat,A,power_power(A,B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_683_nat__mult__le__cancel__disj,axiom,
    ! [K: 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),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_684_mult__le__cancel2,axiom,
    ! [Mb: nat,K: 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),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,B2),Xb)),aa(nat,A,power_power(A,B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Y) ) ) ) ).

% power_increasing_iff
tff(fact_694_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_695_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_696_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_697_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_698_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_699_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_700_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_701_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_702_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_703_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_704_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_705_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_706_mult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.left_commute
tff(fact_707_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_708_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xb: A] :
          ( ( one_one(A) = Xb )
        <=> ( Xb = one_one(A) ) ) ) ).

% one_reorient
tff(fact_709_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_710_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_711_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_712_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
    <=> ( ( K = zero_zero(nat) )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_713_left__add__mult__distrib,axiom,
    ! [Ia: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_714_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
         => ( aa(A,$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),C2)),A2) ) ) ) ).

% mult_left_le
tff(fact_715_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_716_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)),Xb) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_717_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb)),Xb) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_718_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_719_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_720_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_721_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_722_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_723_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_724_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_725_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D2 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_726_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W2: A,Y: 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),Y)),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),Y)) )
        <=> ( ( W2 = Xb )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_727_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),E2)),C2) ) ).

% combine_common_factor
tff(fact_728_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% distrib_right
tff(fact_729_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% distrib_left
tff(fact_730_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_731_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_732_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_733_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Z: A,W2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,Xb,Y)),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),Y),W2)) ) ).

% times_divide_times_eq
tff(fact_734_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Z: A,W2: A] : divide_divide(A,divide_divide(A,Xb,Y),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),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_735_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,divide_divide(A,A2,B2),C2) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_736_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_737_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_738_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_739_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_740_mult__le__mono2,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ia)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_741_mult__le__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_742_mult__le__mono,axiom,
    ! [Ia: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_743_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_744_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_745_add__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),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),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ).

% add_mult_distrib2
tff(fact_746_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)) ).

% add_mult_distrib
tff(fact_747_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_748_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_749_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_750_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_751_less__mult__imp__div__less,axiom,
    ! [Mb: nat,Ia: 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),Ia),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Mb,Nb)),Ia) ) ).

% less_mult_imp_div_less
tff(fact_752_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_753_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_754_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [Z2: A] :
              ( aa(A,$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,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Xb)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_755_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_756_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_757_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_758_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_759_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_760_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_761_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_762_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_763_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xb: A,A2: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_764_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_765_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xb: A,A2: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_766_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_767_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_768_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_769_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_770_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_771_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_772_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_773_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_774_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono
tff(fact_775_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_776_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono
tff(fact_777_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_778_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_779_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_780_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_781_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_782_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_783_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_784_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_785_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_786_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_787_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_788_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_789_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_790_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_791_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C2: A,D2: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_803_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( divide_divide(A,Xb,Y) = 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),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_804_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( divide_divide(A,B2,C2) = A2 )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_805_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = divide_divide(A,B2,C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_806_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
           => ( divide_divide(A,B2,C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_807_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 )
           => ( A2 = divide_divide(A,B2,C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_808_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( divide_divide(A,B2,C2) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_809_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = divide_divide(A,B2,C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_810_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A2) = A2 ) ).

% mult_numeral_1
tff(fact_811_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% mult_numeral_1_right
tff(fact_812_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_813_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_814_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_815_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_816_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = divide_divide(nat,Mb,Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_817_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_818_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_819_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_820_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_821_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_822_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_823_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_824_mult__less__mono1,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ) ).

% mult_less_mono1
tff(fact_825_mult__less__mono2,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Ia)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ) ).

% mult_less_mono2
tff(fact_826_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_827_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_828_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_829_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_830_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_831_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_832_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_833_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_834_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_835_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_836_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_837_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_838_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_839_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_840_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_841_split__div,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,divide_divide(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$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,P,I4) ) ) ) ) ) ).

% split_div
tff(fact_842_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_843_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_844_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_845_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_846_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_847_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_848_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_849_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_850_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_851_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_852_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_853_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_854_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_855_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_856_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : aa(A,$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),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_857_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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),Y),Y))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_858_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : ~ aa(A,$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),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_859_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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),Y),Y)))
        <=> ( ( Xb != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_860_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => ( aa(A,$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,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_861_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => ( aa(A,$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,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_862_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),divide_divide(A,Xb,Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_863_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_864_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_865_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_866_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_867_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_868_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_869_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_870_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W2) = divide_divide(A,B2,C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) = B2,aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_871_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( divide_divide(A,B2,C2) = aa(num,A,numeral_numeral(A),W2) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2),aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_872_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Z)),Y) = 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),Y),Z)),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_873_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,Y,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),Y),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_874_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,Xb,Y)) = 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),Y)),Y) ) ) ) ).

% add_num_frac
tff(fact_875_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Y)),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),Y)),Y) ) ) ) ).

% add_frac_num
tff(fact_876_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,Y)),divide_divide(A,W2,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => ( aa(A,$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,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_890_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),divide_divide(A,Xb,Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_891_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_892_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_893_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_894_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_895_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_896_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => ( aa(A,$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,C2,A2)),divide_divide(A,C2,B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_897_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_898_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_899_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),aa(A,$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_900_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),aa(A,$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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B2,C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),aa(A,$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_913_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),B2),aa(A,$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_914_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),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,Y),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)),Y)) ) ).

% power2_sum
tff(fact_915_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_916_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_917_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K: 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,K)),divide_divide(nat,Nb,K)) ) ).

% div_le_mono
tff(fact_918_ex__power__ivl1,axiom,
    ! [B2: nat,K: 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)),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_919_ex__power__ivl2,axiom,
    ! [B2: nat,K: 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))),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_920_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_921_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_922_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K: 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,K,Nb)),divide_divide(nat,K,Mb)) ) ) ).

% div_le_mono2
tff(fact_923_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xb: A,Y: A] :
          ( ( aa(nat,A,power_power(A,U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => aa(A,$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),Y),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_924_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] : aa(A,$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)),Y)),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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_squares_bound
tff(fact_925_nat__induct2,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,P,N)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct2
tff(fact_926_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_927_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) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Mb))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,Nb) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(X3,Nb)) )
                                   => ( aa(nat,$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),product_Pair(nat,nat,Mia),Maa)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_928_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_929_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_930_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_931_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.bounded_iff
tff(fact_932_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_933_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)
     => ~ ! [N: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) ) ).

% even_odd_cases
tff(fact_934_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_935_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_936_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_937_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_938_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_939_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_940_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_941_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_942_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_943_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_944_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se5668285175392031749et_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_957_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_958_num__double,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),Nb) = bit0(Nb) ).

% num_double
tff(fact_959_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_960_Suc__numeral,axiom,
    ! [Nb: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ).

% Suc_numeral
tff(fact_961_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_962_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_963_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_964_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int] :
            ( aa(int,$o,P,X3)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X: int] :
              ( aa(int,$o,P,X)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_965_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_966_int__gr__induct,axiom,
    ! [K: int,Ia: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),Ia)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,Ia) ) ) ) ).

% int_gr_induct
tff(fact_967_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_968_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_969_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_970_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_971_zmult__zless__mono2,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ia),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ia)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_972_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_973_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_974_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_975_times__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_976_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_977_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa(nat,nat,suc,Nb) ).

% n_not_Suc_n
tff(fact_978_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xb) = aa(nat,nat,suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_979_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_980_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_981_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_982_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% not0_implies_Suc
tff(fact_983_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_not_Suc
tff(fact_984_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_neq_Suc
tff(fact_985_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa(nat,nat,suc,Mb) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_986_zero__induct,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_987_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [X3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X3),zero_zero(nat))
     => ( ! [Y3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y3))
       => ( ! [X3: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X3),Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_988_nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P,N)
           => aa(nat,$o,P,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P,Nb) ) ) ).

% nat_induct
tff(fact_989_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_990_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_991_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_992_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_993_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_994_Nat_OlessE,axiom,
    ! [Ia: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),K)
     => ( ( K != aa(nat,nat,suc,Ia) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_995_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_996_Suc__lessE,axiom,
    ! [Ia: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ia)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_997_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_998_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_999_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_1000_Ex__less__Suc,axiom,
    ! [Nb: nat,P: 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,P,I4) )
    <=> ( aa(nat,$o,P,Nb)
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
            & aa(nat,$o,P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_1001_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_1002_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_1003_All__less__Suc,axiom,
    ! [Nb: nat,P: 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,P,I4) )
    <=> ( aa(nat,$o,P,Nb)
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
           => aa(nat,$o,P,I4) ) ) ) ).

% All_less_Suc
tff(fact_1004_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_1005_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_1006_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_1007_less__trans__Suc,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ia)),K) ) ) ).

% less_trans_Suc
tff(fact_1008_less__Suc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K3)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K3)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K3) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,Ia),J) ) ) ) ).

% less_Suc_induct
tff(fact_1009_strict__inc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I2))
               => aa(nat,$o,P,I2) ) )
         => aa(nat,$o,P,Ia) ) ) ) ).

% strict_inc_induct
tff(fact_1010_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_1011_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_1012_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_1013_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_1014_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)
     => ? [M2: nat] : M6 = aa(nat,nat,suc,M2) ) ).

% Suc_le_D
tff(fact_1015_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_1016_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_1017_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_1018_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),N)
             => aa(nat,$o,P,M) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% full_nat_induct
tff(fact_1019_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,P,Mb)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_1020_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,$o),R,X3),X3)
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X3),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y3),Z2)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X3),Z2) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1021_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A2) )
     => ( aa(nat,nat,suc,A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_1022_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_1023_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_1024_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_1025_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N5: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$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,F2,Nb)),aa(nat,A,F2,N5)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_1026_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,Mb: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1027_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N5: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$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,F2,Nb)),aa(nat,A,F2,N5)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_1028_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N5: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,N))
         => ( aa(nat,$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,F2,N5)),aa(nat,A,F2,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_1029_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_1030_gr0__implies__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_implies_Suc
tff(fact_1031_All__less__Suc2,axiom,
    ! [Nb: nat,P: 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,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_1032_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ? [M3: nat] : Nb = aa(nat,nat,suc,M3) ) ).

% gr0_conv_Suc
tff(fact_1033_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: 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,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
            & aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_1034_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_1035_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_1036_dec__induct,axiom,
    ! [Ia: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(nat,$o,P,Ia)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),J)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) )
         => aa(nat,$o,P,J) ) ) ) ).

% dec_induct
tff(fact_1037_inc__induct,axiom,
    ! [Ia: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),J)
               => ( aa(nat,$o,P,aa(nat,nat,suc,N))
                 => aa(nat,$o,P,N) ) ) )
         => aa(nat,$o,P,Ia) ) ) ) ).

% inc_induct
tff(fact_1038_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_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ~ ! [Q4: nat] : Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q4)) ) ).

% less_natE
tff(fact_1046_less__add__Suc1,axiom,
    ! [Ia: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),Mb))) ).

% less_add_Suc1
tff(fact_1047_less__add__Suc2,axiom,
    ! [Ia: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Ia))) ).

% less_add_Suc2
tff(fact_1048_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
    <=> ? [K2: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)) ) ).

% less_iff_Suc_add
tff(fact_1049_less__imp__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_imp_Suc_add
tff(fact_1050_Suc__mult__less__cancel1,axiom,
    ! [K: 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,K)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_1051_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_1052_Suc__mult__le__cancel1,axiom,
    ! [K: 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,K)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_1053_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_1054_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_1055_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_1056_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1057_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_1058_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_1059_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_1060_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_1061_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_1062_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Nb)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),K3)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,aa(nat,nat,suc,K3)) ) ) ) ).

% ex_least_nat_less
tff(fact_1063_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_1064_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_1065_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_1066_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_1067_power__gt__expt,axiom,
    ! [Nb: nat,K: 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),K),aa(nat,nat,power_power(nat,Nb),K)) ) ).

% power_gt_expt
tff(fact_1068_nat__one__le__power,axiom,
    ! [Ia: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Ia)
     => 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,Ia),Nb)) ) ).

% nat_one_le_power
tff(fact_1069_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_1070_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_1071_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_1072_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_1073_Suc__nat__number__of__add,axiom,
    ! [V2: num,Nb: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),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),V2),one2))),Nb) ).

% Suc_nat_number_of_add
tff(fact_1074_num_Osize_I5_J,axiom,
    ! [X22: num] : aa(num,nat,size_size(num),bit0(X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_1075_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_1076_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_1077_split__div_H,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,divide_divide(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q5: nat] :
            ( aa(nat,$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,P,Q5) ) ) ) ).

% split_div'
tff(fact_1078_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P,N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) )
       => ( ! [N: nat] :
              ( aa(nat,$o,P,N)
             => aa(nat,$o,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_1079_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_1080_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_1081_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_1082_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 )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2)
                  & ( aa(nat,real,power_power(real,Y2),Nb) = A2 ) )
               => ( Y2 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1083_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_1084_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_1085_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_1086_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.boundedE
tff(fact_1087_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2) ) ) ) ).

% max.boundedI
tff(fact_1088_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_1089_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_1090_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_1091_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y))
        <=> ( aa(A,$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),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_1092_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_1093_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_1094_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_1095_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_1096_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_1097_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% max.strict_boundedE
tff(fact_1098_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_1099_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_1100_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_1101_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_1102_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_1103_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_1104_vebt__member_Osimps_I4_J,axiom,
    ! [V2: 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_1105_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_1106_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_1107_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_1108_vebt__member_Osimps_I3_J,axiom,
    ! [V2: 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)),V2),zero_zero(nat),Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_1109_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_1110_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1111_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_1112_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_1113_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_1114_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_1115_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2638667681897837118et_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_1116_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,Xb),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1117_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(nat,real,power_power(real,Xb),N)) ) ).

% real_arch_pow
tff(fact_1118_int__ge__induct,axiom,
    ! [K: int,Ia: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),Ia)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,Ia) ) ) ) ).

% int_ge_induct
tff(fact_1119_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_1120_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_1121_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_1122_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_1123_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Nb: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M),N)
             => aa(extended_enat,$o,P,M) )
         => aa(extended_enat,$o,P,N) )
     => aa(extended_enat,$o,P,Nb) ) ).

% enat_less_induct
tff(fact_1124_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_1125_q__pos__lemma,axiom,
    ! [B5: 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),B5),Q6)),R4))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_1126_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B5: 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),B5),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),B5),Q6)),R4))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B5)
         => ( 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)),B5)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1127_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B5: 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),B5),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),B5),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)),B5)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q6),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1128_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_1129_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_1130_split__zdiv,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,divide_divide(int,Nb,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_1131_zdiv__mono1,axiom,
    ! [A2: int,A5: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A5)
     => ( 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,A5,B2)) ) ) ).

% zdiv_mono1
tff(fact_1132_zdiv__mono2,axiom,
    ! [A2: int,B5: 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)),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B2)),divide_divide(int,A2,B5)) ) ) ) ).

% zdiv_mono2
tff(fact_1133_zdiv__eq__0__iff,axiom,
    ! [Ia: int,K: int] :
      ( ( divide_divide(int,Ia,K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ia)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ia),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),Ia) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1134_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_1135_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_1136_zdiv__mono1__neg,axiom,
    ! [A2: int,A5: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A5)
     => ( 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,A5,B2)),divide_divide(int,A2,B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_1137_zdiv__mono2__neg,axiom,
    ! [A2: int,B5: 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)),B5)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B5),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A2,B5)),divide_divide(int,A2,B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1138_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,K,L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_1139_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),divide_divide(int,K,L)) ) ) ).

% div_positive_int
tff(fact_1140_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_1141_int__div__less__self,axiom,
    ! [Xb: int,K: 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)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,Xb,K)),Xb) ) ) ).

% int_div_less_self
tff(fact_1142_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_1143_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_1144_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_1145_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_1146_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,Ia: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),divide_divide(int,Ia,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),Ia) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1154_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1155_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),product_Pair(A,A,Q2),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_1156_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_1157_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_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1158_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% dbl_simps(3)
tff(fact_1159_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_1160_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A7: $o,B7: $o] : Ta = vEBT_Leaf((A7),(B7)) ) ).

% deg1Leaf
tff(fact_1161_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_1162_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_1163_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_1164_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_1165_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_1166_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_1167_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_1168_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_1169_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_1170_dbl__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,zero_zero(A)) = zero_zero(A) ) ) ).

% dbl_simps(2)
tff(fact_1171_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_1172_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_1173_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_1174_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_1175_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_1176_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_1177_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self4
tff(fact_1178_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self3
tff(fact_1179_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self2
tff(fact_1180_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self1
tff(fact_1181_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_1182_dbl__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bit0(K)) ) ).

% dbl_simps(5)
tff(fact_1183_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: 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),K)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self4
tff(fact_1184_Suc__mod__mult__self3,axiom,
    ! [K: 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),K),Nb)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self3
tff(fact_1185_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K: 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),K))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self2
tff(fact_1186_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K: 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),K),Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self1
tff(fact_1187_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_1188_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_1189_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_1190_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_1191_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_1192_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_1193_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_1194_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_1195_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_eq
tff(fact_1196_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A5: A,B2: A,B5: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A5,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B5,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),B5),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1197_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_left_eq
tff(fact_1198_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) ) ).

% mod_add_right_eq
tff(fact_1199_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_1200_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F3: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc: A] : Xb != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A4),aa(A,product_prod(nat,A),product_Pair(nat,A,B4),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1201_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_1202_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1203_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1204_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1205_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),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1206_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),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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),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_1207_vebt__insert_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xb != aa(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),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),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_1208_vebt__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xb != aa(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),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),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_1209_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [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)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1210_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_1211_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_1212_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_1213_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D4)) ) ) ).

% mod_eqE
tff(fact_1214_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2)) ) ).

% div_add1_eq
tff(fact_1215_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_1216_mod__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat,P3: nat,Mb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),P3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),P3)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),P3)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,modulo_modulo(nat,aa(nat,nat,suc,N),P3)) ) )
           => aa(nat,$o,P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_1217_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [M2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M2),zero_zero(nat))
     => ( ! [M2: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N),modulo_modulo(nat,M2,N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M2),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_1218_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_1219_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_1220_mod__eq__0D,axiom,
    ! [Mb: nat,D2: nat] :
      ( ( modulo_modulo(nat,Mb,D2) = zero_zero(nat) )
     => ? [Q4: nat] : Mb = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q4) ) ).

% mod_eq_0D
tff(fact_1221_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Y,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),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1222_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_1223_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ).

% dbl_def
tff(fact_1224_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv)),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_1225_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_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_1237_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,A2,B2))),modulo_modulo(A,A2,B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_1238_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2)),C2)) ) ).

% div_mult1_eq
tff(fact_1239_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_1240_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_1241_div__less__mono,axiom,
    ! [A3: nat,B3: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( modulo_modulo(nat,A3,Nb) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B3,Nb) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,A3,Nb)),divide_divide(nat,B3,Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_1242_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_1243_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_1244_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Y,Nb) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xb)
       => ? [Q4: nat] : Xb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1245_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_1246_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_1247_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_1248_div__mod__decomp,axiom,
    ! [A3: nat,Nb: nat] : A3 = 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,A3,Nb)),Nb)),modulo_modulo(nat,A3,Nb)) ).

% div_mod_decomp
tff(fact_1249_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_1250_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_1251_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_1252_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_1253_split__mod,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,modulo_modulo(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,Mb) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$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,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_1254_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),modulo_modulo(A,divide_divide(A,A2,B2),C2))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1255_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_1256_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: 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,V2)),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),product_Pair(nat,nat,Xb),Xb)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_1257_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_1258_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_1259_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_1260_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => ( 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,A3,Nb)),
                $ite(modulo_modulo(nat,B3,Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            divide_divide(nat,B3,Nb)) ) ) ).

% verit_le_mono_div
tff(fact_1261_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_1262_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_1263_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_1264_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_1265_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_1266_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: $o,B4: $o] : A1 = vEBT_Leaf((A4),(B4))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => vEBT_invar_vebt(X,N) )
                 => ( vEBT_invar_vebt(Summary,M2)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                     => ( ( M2 = N )
                       => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                         => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => vEBT_invar_vebt(X,N) )
                   => ( vEBT_invar_vebt(Summary,M2)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                       => ( ( M2 = aa(nat,nat,suc,N) )
                         => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                           => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_invar_vebt(X,N) )
                     => ( vEBT_invar_vebt(Summary,M2)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                         => ( ( M2 = N )
                           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                             => ( ! [I3: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_13)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I3) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) )
                                 => ( 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 )
                                         => ! [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))),M2))
                                             => ( ( ( vEBT_VEBT_high(Ma,N) = I3 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(Ma,N)) )
                                                & ! [X: nat] :
                                                    ( ( ( vEBT_VEBT_high(X,N) = I3 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(X,N)) )
                                                   => ( 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),N: nat,Summary: vEBT_VEBT,M2: 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),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,N) )
                       => ( vEBT_invar_vebt(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) )
                           => ( ( M2 = aa(nat,nat,suc,N) )
                             => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M2) )
                               => ( ! [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))),M2))
                                     => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),X_13)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I3) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                         => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I3: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                               => ( ( ( vEBT_VEBT_high(Ma,N) = I3 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(Ma,N)) )
                                                  & ! [X: nat] :
                                                      ( ( ( vEBT_VEBT_high(X,N) = I3 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I3)),vEBT_VEBT_low(X,N)) )
                                                     => ( aa(nat,$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_1267_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A7: $o,B7: $o] : A1 = vEBT_Leaf((A7),(B7))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X2: vEBT_VEBT] :
                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X2,N2) )
            & vEBT_invar_vebt(Summary2,N2)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X2: vEBT_VEBT] :
                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) ) )
        | ? [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X2: vEBT_VEBT] :
                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X2,N2) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N2))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X2: vEBT_VEBT] :
                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) ) )
        | ? [TreeList2: list(vEBT_VEBT),N2: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X2: vEBT_VEBT] :
                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X2,N2) )
            & vEBT_invar_vebt(Summary2,N2)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X2: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),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))),N2))
                 => ( ( ( vEBT_VEBT_high(Ma2,N2) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N2)) )
                    & ! [X2: nat] :
                        ( ( ( vEBT_VEBT_high(X2,N2) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X2,N2)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X2)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list(vEBT_VEBT),N2: 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),product_Pair(nat,nat,Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X2: vEBT_VEBT] :
                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X2,N2) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N2))
            & ( 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,N2)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) )
            & ! [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,N2)))
               => ( ? [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 )
             => ! [X2: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) ) )
            & aa(nat,$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,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N2)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N2) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N2)) )
                    & ! [X2: nat] :
                        ( ( ( vEBT_VEBT_high(X2,N2) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X2,N2)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X2)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1268_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% mult_less_iff1
tff(fact_1269_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_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_1270_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_1271_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),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_1272_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_1273_option_Osize_I4_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_1274_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( aa(nat,$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_1275_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( aa(nat,$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_1276_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_1277_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($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) )
             => ( Y != 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) )
               => ( Y != 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) )
                 => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),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),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                   => ( Y != $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),
                                    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),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_1278_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(bit0(X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_1279_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect(nat,vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_1280_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( dvd_dvd(nat,Mb,one_one(nat))
    <=> ( Mb = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_1281_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,finite_finite2(A),collect(A,P))
        & aa(set(A),$o,finite_finite2(A),collect(A,Q)) ) ) ).

% finite_Collect_disjI
tff(fact_1282_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( aa(set(A),$o,finite_finite2(A),collect(A,P))
        | aa(set(A),$o,finite_finite2(A),collect(A,Q)) )
     => aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_1283_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_ac(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_1284_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_ad(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_1285_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( dvd_dvd(A,zero_zero(A),A2)
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1286_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : dvd_dvd(A,A2,zero_zero(A)) ) ).

% dvd_0_right
tff(fact_1287_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1288_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1289_dvd__1__left,axiom,
    ! [K: nat] : dvd_dvd(nat,aa(nat,nat,suc,zero_zero(nat)),K) ).

% dvd_1_left
tff(fact_1290_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( 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_1291_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,A2,C2)
           => ( dvd_dvd(A,divide_divide(A,B2,A2),divide_divide(A,C2,A2))
            <=> dvd_dvd(A,B2,C2) ) ) ) ) ).

% div_dvd_div
tff(fact_1292_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( ( K = zero_zero(nat) )
        | dvd_dvd(nat,Mb,Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1293_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_1294_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_1295_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(set(A)),$o,finite_finite2(set(A)),collect(set(A),aTP_Lamp_ae(set(A),fun(set(A),$o),A3))) ) ).

% finite_Collect_subsets
tff(fact_1296_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_af(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_1297_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_ag(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_1298_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_1299_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),K))) ).

% finite_Collect_le_nat
tff(fact_1300_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( C2 = zero_zero(A) )
            | dvd_dvd(A,A2,B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1301_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | dvd_dvd(A,A2,B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1302_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))
          <=> dvd_dvd(A,B2,C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1303_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2))
          <=> dvd_dvd(A,B2,C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1304_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_1305_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1306_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),B2))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1307_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( 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_1308_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( 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_1309_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,divide_divide(A,A2,B2),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1310_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => dvd_dvd(A,divide_divide(A,one_one(A),A2),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_1311_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( 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_1312_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,A2)
         => ( dvd_dvd(A,C2,B2)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ) ).

% div_add
tff(fact_1313_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1314_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1315_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1316_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_1317_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( 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_1318_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( 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_1319_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)
         => ( dvd_dvd(A,aa(nat,A,power_power(A,A2),Nb),aa(nat,A,power_power(A,B2),Nb))
          <=> dvd_dvd(A,A2,B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1320_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
          <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),B2) ) ) ) ).

% even_add
tff(fact_1321_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ~ ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),B2) ) ) ) ).

% odd_add
tff(fact_1322_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W2)))
        <=> ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(num,nat,numeral_numeral(nat),W2))
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(num,nat,numeral_numeral(nat),W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1323_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),Nb)),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_1324_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(num,nat,numeral_numeral(nat),W2))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1325_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( 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)))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ).

% even_plus_one_iff
tff(fact_1326_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W2)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
            | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(num,nat,numeral_numeral(nat),W2))
              & ( A2 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(num,nat,numeral_numeral(nat),W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1327_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( 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_1328_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( 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_1329_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ 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_1330_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(nat,A,power_power(A,A2),Nb))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% even_power
tff(fact_1331_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ 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_1332_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,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2))
            & ( ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(num,nat,numeral_numeral(nat),W2))
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( 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_1333_finite__divisors__int,axiom,
    ! [Ia: int] :
      ( ( Ia != zero_zero(int) )
     => aa(set(int),$o,finite_finite2(int),collect(int,aTP_Lamp_aj(int,fun(int,$o),Ia))) ) ).

% finite_divisors_int
tff(fact_1334_Collect__restrict,axiom,
    ! [A: $tType,X5: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),X5),P))),X5) ).

% Collect_restrict
tff(fact_1335_prop__restrict,axiom,
    ! [A: $tType,Xb: A,Z4: set(A),X5: set(A),P: fun(A,$o)] :
      ( member(A,Xb,Z4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z4),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),X5),P)))
       => aa(A,$o,P,Xb) ) ) ).

% prop_restrict
tff(fact_1336_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_al(A,fun(A,$o),A2))),collect(A,aTP_Lamp_al(A,fun(A,$o),B2)))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% subset_divisors_dvd
tff(fact_1337_Collect__subset,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))),A3) ).

% Collect_subset
tff(fact_1338_less__eq__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3)) ) ).

% less_eq_set_def
tff(fact_1339_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : dvd_dvd(A,A2,A2) ) ).

% dvd_refl
tff(fact_1340_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,B2,C2)
           => dvd_dvd(A,A2,C2) ) ) ) ).

% dvd_trans
tff(fact_1341_empty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_am(A,$o)) ).

% empty_def
tff(fact_1342_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),collect(A,P))
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% not_finite_existsD
tff(fact_1343_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),R: fun(A,fun(B,$o))] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( ! [X3: A] :
              ( member(A,X3,A3)
             => ? [Xa: B] :
                  ( member(B,Xa,B3)
                  & aa(B,$o,aa(A,fun(B,$o),R,X3),Xa) ) )
         => ? [X3: B] :
              ( member(B,X3,B3)
              & ~ aa(set(A),$o,finite_finite2(A),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A3),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1344_less__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3)) ) ).

% less_set_def
tff(fact_1345_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),$o),ord_less(set(A)),collect(A,aTP_Lamp_al(A,fun(A,$o),A2))),collect(A,aTP_Lamp_al(A,fun(A,$o),B2)))
        <=> ( dvd_dvd(A,A2,B2)
            & ~ dvd_dvd(A,B2,A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1346_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ao(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_1347_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ap(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_1348_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_1349_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => aa(set(nat),$o,finite_finite2(nat),collect(nat,aTP_Lamp_aq(nat,fun(nat,$o),Mb))) ) ).

% finite_divisors_nat
tff(fact_1350_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),Ia: nat] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ar(fun(nat,$o),fun(nat,fun(nat,$o)),P),Ia))) ).

% finite_M_bounded_by_nat
tff(fact_1351_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,F2,N))
     => aa(set(nat),$o,finite_finite2(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_as(fun(nat,nat),fun(nat,fun(nat,$o)),F2),U))) ) ).

% finite_less_ub
tff(fact_1352_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( dvd_dvd(A,zero_zero(A),A2)
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1353_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_1354_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ~ ! [K3: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvdE
tff(fact_1355_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A2: A,B2: A,K: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => dvd_dvd(A,B2,A2) ) ) ).

% dvdI
tff(fact_1356_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
        <=> ? [K2: A] : A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvd_def
tff(fact_1357_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P3: A,A2: A,B2: A] :
          ( dvd_dvd(A,P3,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ~ ! [X3: A,Y3: A] :
                ( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X3),Y3) )
               => ( dvd_dvd(A,X3,A2)
                 => ~ dvd_dvd(A,Y3,B2) ) ) ) ) ).

% dvd_productE
tff(fact_1358_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C2: A,B2: A] :
          ( dvd_dvd(A,A2,C2)
         => dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult
tff(fact_1359_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ? [B8: A,C6: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B8),C6) )
              & dvd_dvd(A,B8,B2)
              & dvd_dvd(A,C6,C2) ) ) ) ).

% division_decomp
tff(fact_1360_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,B2)
         => dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult2
tff(fact_1361_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2)
         => dvd_dvd(A,A2,C2) ) ) ).

% dvd_mult_left
tff(fact_1362_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% dvd_triv_left
tff(fact_1363_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,C2,D2)
           => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ).

% mult_dvd_mono
tff(fact_1364_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2)
         => dvd_dvd(A,B2,C2) ) ) ).

% dvd_mult_right
tff(fact_1365_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) ) ).

% dvd_triv_right
tff(fact_1366_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : dvd_dvd(A,one_one(A),A2) ) ).

% one_dvd
tff(fact_1367_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => dvd_dvd(A,B2,A2) ) ) ).

% unit_imp_dvd
tff(fact_1368_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,A2,one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1369_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> dvd_dvd(A,A2,C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_1370_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( dvd_dvd(A,A2,C2)
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> dvd_dvd(A,A2,B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_1371_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,A2,C2)
           => dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ) ).

% dvd_add
tff(fact_1372_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,A2)
         => ( dvd_dvd(A,C2,B2)
           => ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1373_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
         => ( dvd_dvd(A,C2,A2)
           => ( dvd_dvd(A,C2,B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1374_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B2: A,A2: A] :
          ( dvd_dvd(A,D2,B2)
         => ( dvd_dvd(A,B2,A2)
           => ( divide_divide(A,divide_divide(A,A2,D2),divide_divide(A,B2,D2)) = divide_divide(A,A2,B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1375_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( dvd_dvd(nat,zero_zero(nat),A2)
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_1376_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( dvd_dvd(nat,A2,zero_zero(nat))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_1377_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( dvd_dvd(nat,zero_zero(nat),A2)
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_1378_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( dvd_dvd(nat,zero_zero(nat),A2)
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_1379_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : dvd_dvd(nat,A2,zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_1380_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A2: A] :
          ( dvd_dvd(A,C2,B2)
         => ( dvd_dvd(A,C2,modulo_modulo(A,A2,B2))
          <=> dvd_dvd(A,C2,A2) ) ) ) ).

% dvd_mod_iff
tff(fact_1381_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,modulo_modulo(A,A2,B2))
         => ( dvd_dvd(A,C2,B2)
           => dvd_dvd(A,C2,A2) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1382_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_1383_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect(nat,vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_1384_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L))
    <=> ( dvd_dvd(int,L,K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_1385_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ dvd_dvd(A,zero_zero(A),one_one(A)) ) ).

% not_is_unit_0
tff(fact_1386_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S))
          <=> dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S)) ) ) ) ).

% pinf(9)
tff(fact_1387_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X)
         => ( ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S))
          <=> ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S)) ) ) ) ).

% pinf(10)
tff(fact_1388_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ( dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S))
          <=> dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S)) ) ) ) ).

% minf(9)
tff(fact_1389_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z2: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z2)
         => ( ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S))
          <=> ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S)) ) ) ) ).

% minf(10)
tff(fact_1390_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1391_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),one_one(A))
        <=> ( dvd_dvd(A,A2,one_one(A))
            & dvd_dvd(A,B2,one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_1392_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> dvd_dvd(A,A2,C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_1393_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2)
          <=> dvd_dvd(A,A2,C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_1394_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> dvd_dvd(A,A2,C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1395_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2)
          <=> dvd_dvd(A,B2,C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1396_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_1397_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_1398_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( dvd_dvd(A,C2,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C2)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C2) ) ) ) ).

% dvd_div_mult
tff(fact_1399_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( dvd_dvd(A,C2,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ) ) ).

% div_mult_swap
tff(fact_1400_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( dvd_dvd(A,C2,B2)
         => ( dvd_dvd(A,B2,A2)
           => ( divide_divide(A,A2,divide_divide(A,B2,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_1401_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2),A2)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_1402_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2)
         => dvd_dvd(A,A2,divide_divide(A,B2,C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_1403_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D2: A,C2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( dvd_dvd(A,D2,C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),divide_divide(A,C2,D2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1404_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( ( divide_divide(A,B2,A2) = divide_divide(A,C2,A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_1405_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,divide_divide(A,A2,B2),C2)
          <=> dvd_dvd(A,A2,C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1406_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A2,divide_divide(A,C2,B2))
          <=> dvd_dvd(A,A2,C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1407_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( dvd_dvd(A,C2,B2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1408_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,A2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1409_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> dvd_dvd(A,B2,A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1410_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1411_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => dvd_dvd(A,B2,A2) ) ) ).

% mod_0_imp_dvd
tff(fact_1412_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)
         => 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_1413_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat,B2: A,Mb: nat] :
          ( dvd_dvd(A,aa(nat,A,power_power(A,A2),Nb),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => dvd_dvd(A,aa(nat,A,power_power(A,A2),Mb),B2) ) ) ) ).

% power_le_dvd
tff(fact_1414_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,Nb: nat,Mb: nat] :
          ( dvd_dvd(A,Xb,Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => dvd_dvd(A,aa(nat,A,power_power(A,Xb),Nb),aa(nat,A,power_power(A,Y),Mb)) ) ) ) ).

% dvd_power_le
tff(fact_1415_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),modulo_modulo(int,K,L)) ) ).

% neg_mod_bound
tff(fact_1416_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,K,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1417_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( 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_1418_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)
       => ~ dvd_dvd(nat,Nb,Mb) ) ) ).

% nat_dvd_not_less
tff(fact_1419_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)
       => ( dvd_dvd(int,Mb,Nb)
         => ( dvd_dvd(int,Nb,Mb)
           => ( Mb = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_1420_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)
       => ~ dvd_dvd(int,Nb,Mb) ) ) ).

% zdvd_not_zless
tff(fact_1421_zdvd__mult__cancel,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),Mb),aa(int,int,aa(int,fun(int,int),times_times(int),K),Nb))
     => ( ( K != zero_zero(int) )
       => dvd_dvd(int,Mb,Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_1422_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,Xb: nat,Y: nat] :
      ( dvd_dvd(nat,D2,A2)
     => ( dvd_dvd(nat,D2,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),Y)),D2) )
            | ( 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),Y)),D2) ) )
         => ? [X3: nat,Y3: nat] :
              ( dvd_dvd(nat,D2,A2)
              & dvd_dvd(nat,D2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Y3)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1423_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D4: nat,X3: nat,Y3: nat] :
      ( dvd_dvd(nat,D4,A2)
      & dvd_dvd(nat,D4,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),Y3)),D4) )
        | ( 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),Y3)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_1424_zdvd__reduce,axiom,
    ! [K: int,Nb: int,Mb: int] :
      ( dvd_dvd(int,K,aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),Mb)))
    <=> dvd_dvd(int,K,Nb) ) ).

% zdvd_reduce
tff(fact_1425_zdvd__period,axiom,
    ! [A2: int,D2: int,Xb: int,Ta: int,C2: int] :
      ( dvd_dvd(int,A2,D2)
     => ( dvd_dvd(int,A2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Ta))
      <=> 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),C2),D2))),Ta)) ) ) ).

% zdvd_period
tff(fact_1426_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_at(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_eq
tff(fact_1427_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),bit_se8732182000553998342ip_bit(A,Mb,A2))
        <=> ~ ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            <=> ( Mb = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_1428_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        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_1429_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_au(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_le
tff(fact_1430_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C4) ) ) ) ).

% unit_dvdE
tff(fact_1431_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X2: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X2))
        <=> ? [X2: A] :
              ( dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),zero_zero(A)))
              & aa(A,$o,P,X2) ) ) ) ).

% unity_coeff_ex
tff(fact_1432_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( dvd_dvd(A,A2,B2)
             => ( dvd_dvd(A,C2,D2)
               => ( ( divide_divide(A,B2,A2) = divide_divide(A,D2,C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1433_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( dvd_dvd(A,C2,B2)
           => ( dvd_dvd(A,A2,divide_divide(A,B2,C2))
            <=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1434_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( dvd_dvd(A,B2,A2)
           => ( dvd_dvd(A,divide_divide(A,A2,B2),C2)
            <=> dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1435_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( dvd_dvd(A,A2,B2)
           => ( ( divide_divide(A,B2,A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1436_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( 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_1437_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,C2,one_one(A))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1438_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_1439_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2) ) ) ) ).

% unit_div_commute
tff(fact_1440_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( dvd_dvd(A,C2,one_one(A))
         => ( dvd_dvd(A,B2,A2)
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = divide_divide(A,divide_divide(A,A2,B2),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_1441_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( A2 = divide_divide(A,C2,B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_1442_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( divide_divide(A,A2,B2) = C2 )
          <=> ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_1443_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1444_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(nat,A,power_power(A,A2),Nb),one_one(A))
        <=> ( dvd_dvd(A,A2,one_one(A))
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1445_zmod__trivial__iff,axiom,
    ! [Ia: int,K: int] :
      ( ( modulo_modulo(int,Ia,K) = Ia )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ia)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ia),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),Ia) ) ) ) ).

% zmod_trivial_iff
tff(fact_1446_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_1447_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_1448_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_1449_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1450_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( dvd_dvd(nat,K,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),K),Nb) ) ) ).

% dvd_imp_le
tff(fact_1451_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => ( ( modulo_modulo(int,A3,Nb) = zero_zero(int) )
         => ( ( modulo_modulo(int,B3,Nb) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A3,Nb)),divide_divide(int,B3,Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1452_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> dvd_dvd(nat,Mb,Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1453_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => dvd_dvd(nat,Mb,Nb) ) ) ).

% dvd_mult_cancel
tff(fact_1454_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D4: nat,X3: nat,Y3: nat] :
          ( dvd_dvd(nat,D4,A2)
          & dvd_dvd(nat,D4,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),Y3)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1455_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( 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_1456_div__mod__decomp__int,axiom,
    ! [A3: int,Nb: int] : A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),divide_divide(int,A3,Nb)),Nb)),modulo_modulo(int,A3,Nb)) ).

% div_mod_decomp_int
tff(fact_1457_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))
    <=> ~ dvd_dvd(nat,Nb,Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1458_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_1459_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Y != $ite(
                      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_1460_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),zero_zero(A)) ) ).

% even_zero
tff(fact_1461_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( 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_1462_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( 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_1463_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( dvd_dvd(A,B4,one_one(A))
                   => ( ( divide_divide(A,one_one(A),A2) = B4 )
                     => ( ( divide_divide(A,one_one(A),B4) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B4) = one_one(A) )
                         => ( divide_divide(A,C2,A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1464_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),B2)
           => 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_1465_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,Mb: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( dvd_dvd(A,aa(nat,A,power_power(A,Xb),Mb),aa(nat,A,power_power(A,Xb),Nb))
          <=> ( dvd_dvd(A,Xb,one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ) ).

% dvd_power_iff
tff(fact_1466_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) ) )
         => dvd_dvd(A,Xb,aa(nat,A,power_power(A,Xb),Nb)) ) ) ).

% dvd_power
tff(fact_1467_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_1468_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1469_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( 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_1470_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( 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_1471_power__dvd__imp__le,axiom,
    ! [Ia: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,aa(nat,nat,power_power(nat,Ia),Mb),aa(nat,nat,power_power(nat,Ia),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Ia)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_1472_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( 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_1473_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
         => ( aa(A,$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_1474_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ 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_1475_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_1476_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_1477_split__zmod,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,modulo_modulo(int,Nb,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,Nb) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_1478_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_1479_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_1480_dvd__power__iff__le,axiom,
    ! [K: 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))),K)
     => ( dvd_dvd(nat,aa(nat,nat,power_power(nat,K),Mb),aa(nat,nat,power_power(nat,K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_1481_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),bit_se2638667681897837118et_bit(A,Mb,A2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            | ( Mb = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_1482_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),bit_se5668285175392031749et_bit(A,Mb,A2))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
            & ( Mb != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_1483_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_1484_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V2: nat,TreeLista: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V2),TreeLista,S),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_1485_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ 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_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( 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) ) )
         => ~ ( ~ 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_1487_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_1488_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
         => aa(A,$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_1489_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
         => ( aa(A,$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_1490_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Nb))
        <=> ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1491_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B3)
     => ( 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,A3,Nb)),
                $ite(modulo_modulo(int,B3,Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            divide_divide(int,B3,Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1492_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,Nb,K)),modulo_modulo(int,Nb,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1493_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,Nb,K)),modulo_modulo(int,Nb,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1494_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeLista: list(vEBT_VEBT),Vd2: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeLista,Vd2),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_1495_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),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_1496_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V2: 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),product_Pair(nat,nat,Mia),Maa)),aa(nat,nat,suc,V2),TreeLista,Vc),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_1497_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),Nb))
        <=> ( ( Nb = zero_zero(nat) )
            | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
              & ( A2 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1498_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $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)
           => (Y) )
         => ~ ! [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)
               => ( (Y)
                <=> ~ $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_1499_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_1500_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_1501_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),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),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_1502_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,$o),ord_less(nat),zero_zero(nat)),Nb)
            & ( ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1503_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),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_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),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),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__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => (Y) )
         => ( ! [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),product_Pair(nat,nat,Mi),Ma)),zero_zero(nat),Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( 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),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2)
                 => ( (Y)
                  <=> ~ ( ( 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)
                   => ( (Y)
                    <=> ~ $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_1506_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),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),
                  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),product_Pair(nat,nat,Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
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),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_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $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)
           => (Y) )
         => ( ( ? [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)
             => (Y) )
           => ( ( ? [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)
               => (Y) )
             => ~ ! [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),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
                   => ( (Y)
                    <=> ~ $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_1509_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(set(complex),$o,finite_finite2(complex),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_av(nat,fun(complex,fun(complex,$o)),Nb),C2))) ) ).

% finite_nth_roots
tff(fact_1510_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),collect(A,aTP_Lamp_aw(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_1511_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($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),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) )
               => ( ( Y = 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),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) )
                 => ( ( Y = 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),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) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),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),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( Y = $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),
                                        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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1512_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_1513_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_1514_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),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),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),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),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),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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_1515_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $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),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) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V3: product_prod(nat,nat),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) )
                 => ( ~ (Y)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),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) )
                   => ( ~ (Y)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),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),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( (Y)
                        <=> $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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1516_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),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),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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_1517_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,Mb,Nb)
     => ( dvd_dvd(nat,Nb,Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_1518_gcd__nat_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
     => ~ ( dvd_dvd(nat,B2,A2)
          & ( B2 != A2 ) ) ) ).

% gcd_nat.asym
tff(fact_1519_gcd__nat_Orefl,axiom,
    ! [A2: nat] : dvd_dvd(nat,A2,A2) ).

% gcd_nat.refl
tff(fact_1520_gcd__nat_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( dvd_dvd(nat,A2,B2)
     => ( dvd_dvd(nat,B2,C2)
       => dvd_dvd(nat,A2,C2) ) ) ).

% gcd_nat.trans
tff(fact_1521_gcd__nat_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( dvd_dvd(nat,A2,B2)
        & dvd_dvd(nat,B2,A2) ) ) ).

% gcd_nat.eq_iff
tff(fact_1522_gcd__nat_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( dvd_dvd(nat,A2,A2)
        & ( A2 != A2 ) ) ).

% gcd_nat.irrefl
tff(fact_1523_gcd__nat_Oantisym,axiom,
    ! [A2: nat,B2: nat] :
      ( dvd_dvd(nat,A2,B2)
     => ( dvd_dvd(nat,B2,A2)
       => ( A2 = B2 ) ) ) ).

% gcd_nat.antisym
tff(fact_1524_gcd__nat_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
     => ( ( dvd_dvd(nat,B2,C2)
          & ( B2 != C2 ) )
       => ( dvd_dvd(nat,A2,C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_1525_gcd__nat_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( dvd_dvd(nat,A2,B2)
     => ( ( dvd_dvd(nat,B2,C2)
          & ( B2 != C2 ) )
       => ( dvd_dvd(nat,A2,C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_1526_gcd__nat_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
     => ( dvd_dvd(nat,B2,C2)
       => ( dvd_dvd(nat,A2,C2)
          & ( A2 != C2 ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_1527_gcd__nat_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
    <=> ( dvd_dvd(nat,A2,B2)
        & ~ dvd_dvd(nat,B2,A2) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_1528_gcd__nat_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( dvd_dvd(nat,A2,B2)
    <=> ( ( dvd_dvd(nat,A2,B2)
          & ( A2 != B2 ) )
        | ( A2 = B2 ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_1529_gcd__nat_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
    <=> ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_1530_gcd__nat_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
     => dvd_dvd(nat,A2,B2) ) ).

% gcd_nat.strict_implies_order
tff(fact_1531_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd(nat,A2,B2)
        & ( A2 != B2 ) )
     => ( A2 != B2 ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_1532_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( dvd_dvd(nat,A2,B2)
       => ( dvd_dvd(nat,A2,B2)
          & ( A2 != B2 ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_1533_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $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),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) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) ) )
           => ~ ! [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) )
                 => ( ( (Y)
                    <=> $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),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S3)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1534_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),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),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),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_1535_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),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),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),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),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_1536_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),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),product_Pair(nat,nat,Mi),Ma)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( 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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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) )
                     => ( ( (Y)
                        <=> $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),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_1537_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),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),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),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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),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_1538_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),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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),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),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),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),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_1539_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1540_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arsinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_1541_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_1542_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = 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)) )
                 => ( ( Y = $ite(
                          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_1543_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X22: A] : size_option(A,Xb,aa(A,option(A),some(A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1544_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),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_1545_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> ( aa(nat,$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)
              & 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_1546_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Nb)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_1547_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_1548_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S2: set(A)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),R)),aTP_Lamp_a(set(A),fun(A,$o),S2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R),S2) ) ).

% pred_subset_eq
tff(fact_1549_verit__minus__simplify_I4_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),B2)) = B2 ) ).

% verit_minus_simplify(4)
tff(fact_1550_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_1551_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_1552_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_1553_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_1554_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_1555_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_1556_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_1557_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_1558_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_1559_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_1560_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_1561_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_1562_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_1563_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_1564_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Mb: num,Nb: num] :
          ( ( 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)) )
        <=> ( Mb = Nb ) ) ) ).

% neg_numeral_eq_iff
tff(fact_1565_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_1566_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_1567_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_1568_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_1569_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_1570_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_1571_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_1572_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_1573_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_1574_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_1575_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_1576_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_1577_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_1578_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] :
          ( dvd_dvd(A,aa(A,A,uminus_uminus(A),Xb),Y)
        <=> dvd_dvd(A,Xb,Y) ) ) ).

% minus_dvd_iff
tff(fact_1579_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] :
          ( dvd_dvd(A,Xb,aa(A,A,uminus_uminus(A),Y))
        <=> dvd_dvd(A,Xb,Y) ) ) ).

% dvd_minus_iff
tff(fact_1580_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K) ).

% Suc_diff_diff
tff(fact_1581_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_1582_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_1583_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_1584_diff__diff__cancel,axiom,
    ! [Ia: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),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),Ia)) = Ia ) ) ).

% diff_diff_cancel
tff(fact_1585_diff__diff__left,axiom,
    ! [Ia: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ia),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1586_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_1587_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_1588_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = zero_zero(A) )
        <=> ~ (P) ) ) ).

% of_bool_eq_0_iff
tff(fact_1589_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( ~ (P)
            & (Q) ) ) ) ).

% of_bool_less_iff
tff(fact_1590_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = one_one(A) )
        <=> (P) ) ) ).

% of_bool_eq_1_iff
tff(fact_1591_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_1592_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            | (Q) )) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_or_iff
tff(fact_1593_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_1594_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_1595_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_1596_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_1597_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_1598_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_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V2: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_1610_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A2: A,B2: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V2))) ) ).

% left_diff_distrib_numeral
tff(fact_1611_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_1612_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_1613_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_1614_mult__minus1__right,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1_right
tff(fact_1615_mult__minus1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),one_one(A))),Z) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1
tff(fact_1616_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_1617_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_1618_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_1619_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( dvd_dvd(A,C2,A2)
         => ( dvd_dvd(A,C2,B2)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ) ).

% div_diff
tff(fact_1620_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P)))
        <=> (P) ) ) ).

% zero_less_of_bool_iff
tff(fact_1621_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_1622_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_1623_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_1624_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A))
        <=> ~ (P) ) ) ).

% of_bool_less_one_iff
tff(fact_1625_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: $o] : aa($o,A,zero_neq_one_of_bool(A),~ (P)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% of_bool_not_iff
tff(fact_1626_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1627_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),Ia) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Ia)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1628_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1629_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_1630_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_1631_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_simps(1)
tff(fact_1632_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_1633_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_1634_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_1635_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( Nb = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_1636_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) )
        <=> ( Nb = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_1637_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_1638_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_1639_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),V2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_1640_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_1641_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_1642_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ia),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1643_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),Ia) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Ia)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1644_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),W2)))),Y) ) ).

% semiring_norm(168)
tff(fact_1645_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(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),Mb),Nb)) ) ).

% diff_numeral_simps(2)
tff(fact_1646_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb))) ) ).

% diff_numeral_simps(3)
tff(fact_1647_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(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,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1648_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1649_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(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(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1650_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_1651_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_1652_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_1653_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_1654_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_1655_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_1656_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_1657_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_1658_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_1659_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_1660_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_1661_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_1662_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_1663_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_1664_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> 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_1665_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_1666_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb)) ) ).

% diff_numeral_special(3)
tff(fact_1667_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),one2))) ) ).

% diff_numeral_special(4)
tff(fact_1668_dbl__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% dbl_simps(4)
tff(fact_1669_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ 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_1670_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( 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)
        | 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_1671_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_1672_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( 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_1673_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_1674_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_1675_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_1676_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_1677_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B3: A,K: A,B2: A,A2: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_1678_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o,Q2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P3)) = aa($o,A,zero_neq_one_of_bool(A),(Q2)) )
        <=> ( (P3)
          <=> (Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_1679_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_1680_diff__commute,axiom,
    ! [Ia: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ia),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Ia),K)),J) ).

% diff_commute
tff(fact_1681_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) ) ).

% diff_right_commute
tff(fact_1682_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_1683_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_1684_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_1685_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( ( A2 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1686_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            & (Q) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_conj
tff(fact_1687_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_mono
tff(fact_1688_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$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),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_left_mono
tff(fact_1689_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_right_mono
tff(fact_1690_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1691_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_1692_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_1693_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_1694_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$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),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_1695_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_1696_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_1697_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_1698_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_1699_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib'
tff(fact_1700_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ).

% left_diff_distrib'
tff(fact_1701_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib
tff(fact_1702_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% left_diff_distrib
tff(fact_1703_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_1704_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_1705_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_1706_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1707_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = C2 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_1708_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_1709_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_1710_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_1711_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_1712_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1713_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A2 )
         => ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_1714_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_1715_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2)) ) ).

% add_diff_add
tff(fact_1716_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Mb: num,Nb: num] : aa(num,A,numeral_numeral(A),Mb) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% numeral_neq_neg_numeral
tff(fact_1717_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Mb: num,Nb: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb)) != aa(num,A,numeral_numeral(A),Nb) ) ).

% neg_numeral_neq_numeral
tff(fact_1718_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_1719_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_1720_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ).

% diff_divide_distrib
tff(fact_1721_one__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( one_one(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% one_neq_neg_one
tff(fact_1722_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_1723_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_1724_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_1725_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_1726_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_1727_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_1728_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( dvd_dvd(A,Xb,Y)
         => ( dvd_dvd(A,Xb,Z)
           => dvd_dvd(A,Xb,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ) ) ).

% dvd_diff
tff(fact_1729_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,Ia: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),Ia)) ) ) ).

% zero_induct_lemma
tff(fact_1730_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_1731_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_1732_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_1733_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_1734_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_1735_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),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),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_1736_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1737_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_1738_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_1739_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_1740_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_1741_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_1742_Nat_Odiff__cancel,axiom,
    ! [K: 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),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% Nat.diff_cancel
tff(fact_1743_diff__cancel2,axiom,
    ! [Mb: nat,K: 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),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% diff_cancel2
tff(fact_1744_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_1745_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_1746_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),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),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ).

% diff_mult_distrib2
tff(fact_1747_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)) ).

% diff_mult_distrib
tff(fact_1748_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),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),Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_1749_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K,Mb)
     => ( dvd_dvd(nat,K,Nb)
       => dvd_dvd(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_1750_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_1751_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_1752_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),divide_divide(A,Xb,Z))),Y) = 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),Y),Z)),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1753_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),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),K)) = 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),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1754_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_1755_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_1756_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P3)) = $ite((P3),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_1757_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ( ( (P3)
             => aa(A,$o,P,one_one(A)) )
            & ( ~ (P3)
             => aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_1758_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ~ ( ( (P3)
                & ~ aa(A,$o,P,one_one(A)) )
              | ( ~ (P3)
                & ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_1759_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_1760_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_1761_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_le_eq
tff(fact_1762_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% le_diff_eq
tff(fact_1763_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_1764_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2)) ) ) ).

% le_add_diff
tff(fact_1765_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1766_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1767_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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),C2),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1768_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1769_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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),C2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1770_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1771_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_1772_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$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) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1773_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ia: A,K: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_1774_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ia: A,K: A,Nb: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1775_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_less_eq
tff(fact_1776_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% less_diff_eq
tff(fact_1777_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_1778_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_1779_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_1780_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_1781_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E2)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_1782_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E2)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_1783_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xb: A,Y: 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),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) ) ).

% square_diff_square_factored
tff(fact_1784_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xb: A,Y: 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),Y)),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),Y),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_1785_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_1786_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_1787_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_1788_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_1789_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_1790_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_1791_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_1792_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_1793_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_1794_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_1795_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_1796_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_1797_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_1798_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_1799_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_neq_neg_numeral
tff(fact_1800_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),Nb) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_1801_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_1802_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( 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_1803_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( 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_1804_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : 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_1805_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_1806_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_1807_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_1808_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_1809_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_1810_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: 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),C2),A2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2)) ) ) ).

% diff_less_mono
tff(fact_1811_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_1812_less__diff__conv,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),J) ) ).

% less_diff_conv
tff(fact_1813_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_1814_le__diff__conv,axiom,
    ! [J: nat,K: nat,Ia: 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),K)),Ia)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)) ) ).

% le_diff_conv
tff(fact_1815_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1816_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1817_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Ia)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),Ia) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1818_Nat_Ole__imp__diff__is__add,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Ia) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Ia) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1819_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_1820_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( 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)
        | dvd_dvd(nat,Mb,Nb) ) ) ).

% dvd_minus_self
tff(fact_1821_dvd__diffD,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( dvd_dvd(nat,K,Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => dvd_dvd(nat,K,Mb) ) ) ) ).

% dvd_diffD
tff(fact_1822_dvd__diffD1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( dvd_dvd(nat,K,Mb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => dvd_dvd(nat,K,Nb) ) ) ) ).

% dvd_diffD1
tff(fact_1823_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( dvd_dvd(nat,Mb,Nb)
      <=> dvd_dvd(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_1824_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D4: nat,X3: nat,Y3: nat] :
      ( dvd_dvd(nat,D4,A2)
      & dvd_dvd(nat,D4,B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = D4 )
        | ( 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),Y3)) = D4 ) ) ) ).

% bezout1_nat
tff(fact_1825_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_1826_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_1827_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_1828_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_1829_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_1830_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_1831_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2))
        <=> aa(A,$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)),E2)),C2)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1832_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E2)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1833_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_1834_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_1835_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2))
        <=> aa(A,$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)),E2)),C2)),D2) ) ) ).

% less_add_iff1
tff(fact_1836_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E2)),D2)) ) ) ).

% less_add_iff2
tff(fact_1837_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_1838_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_1839_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_1840_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_1841_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_1842_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Xb,Y)),divide_divide(A,W2,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1843_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,Y,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),Y),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1844_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Xb,Z)),Y) = 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),Y),Z)),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1845_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_1846_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_1847_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_1848_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_1849_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_1850_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_1851_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_1852_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_1853_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_1854_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_1855_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_1856_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_1857_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_1858_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,uminus_uminus(A),B2),A2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_1859_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)) = A2 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_1860_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_1861_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_1862_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_1863_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(1)
tff(fact_1864_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(2)
tff(fact_1865_uminus__numeral__One,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% uminus_numeral_One
tff(fact_1866_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D5: A,Ta: A] :
          ( dvd_dvd(A,D2,D5)
         => ! [X: A,K4: A] :
              ( ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Ta))
            <=> ~ dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_1867_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D5: A,Ta: A] :
          ( dvd_dvd(A,D2,D5)
         => ! [X: A,K4: A] :
              ( dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Ta))
            <=> dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_1868_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_1869_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_1870_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_1871_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_1872_diff__Suc__less,axiom,
    ! [Nb: nat,Ia: 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,Ia))),Nb) ) ).

% diff_Suc_less
tff(fact_1873_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,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,$o,P,zero_zero(nat)) )
        & ! [D6: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
           => aa(nat,$o,P,D6) ) ) ) ).

% nat_diff_split
tff(fact_1874_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,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,$o,P,zero_zero(nat)) )
          | ? [D6: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
              & ~ aa(nat,$o,P,D6) ) ) ) ).

% nat_diff_split_asm
tff(fact_1875_less__diff__conv2,axiom,
    ! [K: nat,J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),Ia)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)) ) ) ).

% less_diff_conv2
tff(fact_1876_nat__eq__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),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),Ia),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_1877_nat__eq__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ia),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),Ia)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1878_nat__le__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia)
     => ( 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),Ia),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),Ia),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_1879_nat__le__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),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),Ia),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),Ia)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_1880_nat__diff__add__eq1,axiom,
    ! [J: nat,Ia: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia)
     => ( 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),Ia),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),Ia),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_1881_nat__diff__add__eq2,axiom,
    ! [Ia: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),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),Ia),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),Ia)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_1882_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) )
      <=> dvd_dvd(nat,Q2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1883_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_1884_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),divide_divide(A,W2,Z))
            <=> aa(A,$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),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_1885_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xb,Y)),divide_divide(A,W2,Z))
            <=> aa(A,$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),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_1886_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_1887_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_1888_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_1889_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_1890_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_1891_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_1892_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W2: num] :
          ( ( divide_divide(A,B2,C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_1893_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,B2,C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_1894_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,Xb,Z))),Y) = 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),Y),Z)),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1895_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_1896_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_1897_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_1898_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_1899_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_1900_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_1901_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_1902_nat__less__add__iff1,axiom,
    ! [J: nat,Ia: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia)
     => ( 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),Ia),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),Ia),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_1903_nat__less__add__iff2,axiom,
    ! [Ia: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),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),Ia),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),Ia)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_1904_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_1905_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))
       => ( dvd_dvd(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Q2))
        <=> 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_1906_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)
       => ( 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_1907_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),B3)
     => ( 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,B3,Nb)),divide_divide(int,A3,Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1908_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_1909_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),product_Pair(A,B,X),Xa),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_1910_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R2: A,S: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S)
             => aa(A,$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),V2),U)),S))),V2) ) ) ) ) ).

% scaling_mono
tff(fact_1911_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_1912_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_1913_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$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),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_1914_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_1915_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_1916_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2)
          <=> aa(A,$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),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_1917_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( aa(A,$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,C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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))),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),aa(A,$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_1918_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),aa(A,$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_1919_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_1920_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_1921_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_1922_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P3: A,Mb: nat] :
          aa(nat,A,power_power(A,P3),Mb) = $ite(Mb = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,power_power(A,P3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1923_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_1924_diff__le__diff__pow,axiom,
    ! [K: 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))),K)
     => 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,K),Mb)),aa(nat,nat,power_power(nat,K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_1925_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_1926_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A4: A] :
              ( ( divide_divide(A,A4,aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
             => aa(A,$o,P,A4) )
         => ( ! [A4: A,B4: $o] :
                ( aa(A,$o,P,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,P,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,P,A2) ) ) ) ).

% bits_induct
tff(fact_1927_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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))),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),B2),aa(A,$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_1928_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C2: A] :
          ( aa(A,$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,C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$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))),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),aa(A,$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_1929_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_1930_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( divide_divide(int,K,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1931_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_1932_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_1933_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),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,Y),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)),Y)) ) ).

% power2_diff
tff(fact_1934_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_1935_int__power__div__base,axiom,
    ! [Mb: nat,K: 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)),K)
       => ( divide_divide(int,aa(nat,int,power_power(int,K),Mb),K) = aa(nat,int,power_power(int,K),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_1936_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_1937_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_1938_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_1939_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_1940_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_1941_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)) )
     => 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_1942_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_1943_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)))
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_1944_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(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),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),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_1945_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_1946_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_1947_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% compl_less_compl_iff
tff(fact_1948_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% compl_le_compl_iff
tff(fact_1949_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,Nb: nat] :
      ( ! [N: nat] : aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,H,N)
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( aa(nat,A,F2,Nb) = $ite(Nb = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_1950_Diff__cancel,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),A3) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1951_empty__Diff,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A3) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1952_Diff__empty,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),bot_bot(set(A))) = A3 ).

% Diff_empty
tff(fact_1953_finite__Diff2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
      <=> aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% finite_Diff2
tff(fact_1954_finite__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) ) ).

% finite_Diff
tff(fact_1955_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3) ) ).

% Compl_subset_Compl_iff
tff(fact_1956_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B3)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) ) ).

% Compl_anti_mono
tff(fact_1957_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% Diff_eq_empty_iff
tff(fact_1958_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,L: A,U: A] :
          ( member(A,Ia,set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Ia)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_1959_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_1960_ln__inj__iff,axiom,
    ! [Xb: real,Y: 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)),Y)
       => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,ln_ln(real),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_1961_ln__less__cancel__iff,axiom,
    ! [Xb: real,Y: 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)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_1962_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or1337092689740270186AtMost(nat,L,U)) ).

% finite_atLeastAtMost
tff(fact_1963_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_1964_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_1965_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1966_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_1967_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_1968_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: 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)),Y)
       => ( 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),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_1969_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_1970_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_1971_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_1972_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_1973_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_1974_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_1975_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_1976_less__by__empty,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( ( A3 = bot_bot(set(product_prod(A,A))) )
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),A3),B3) ) ).

% less_by_empty
tff(fact_1977_bot2E,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),Xb),Y) ).

% bot2E
tff(fact_1978_ln__div,axiom,
    ! [Xb: real,Y: 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)),Y)
       => ( aa(real,real,ln_ln(real),divide_divide(real,Xb,Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_1979_Diff__infinite__finite,axiom,
    ! [A: $tType,T3: set(A),S2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),T3)
     => ( ~ 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),T3)) ) ) ).

% Diff_infinite_finite
tff(fact_1980_double__diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1981_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),$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)),A3),B3)),A3) ).

% Diff_subset
tff(fact_1982_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),D5: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D5),B3)
       => aa(set(A),$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)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),D5)) ) ) ).

% Diff_mono
tff(fact_1983_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_1984_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_1985_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_1986_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_1987_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
     => ? [B4: A] : member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_1988_zdvd__zdiffD,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( dvd_dvd(int,K,aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),Nb))
     => ( dvd_dvd(int,K,Nb)
       => dvd_dvd(int,K,Mb) ) ) ).

% zdvd_zdiffD
tff(fact_1989_ln__diff__le,axiom,
    ! [Xb: real,Y: 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)),Y)
       => 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),Y))),divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y),Y)) ) ) ).

% ln_diff_le
tff(fact_1990_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_1991_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_1992_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_1993_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_1994_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_1995_ex__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),Nb)
          & aa(nat,$o,P,M3) )
    <=> ? [X2: nat] :
          ( member(nat,X2,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X2) ) ) ).

% ex_nat_less
tff(fact_1996_all__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),Nb)
         => aa(nat,$o,P,M3) )
    <=> ! [X2: nat] :
          ( member(nat,X2,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X2) ) ) ).

% all_nat_less
tff(fact_1997_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_1998_real__add__less__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_less_0_iff
tff(fact_1999_real__0__less__add__iff,axiom,
    ! [Xb: real,Y: 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),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_2000_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_2001_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_2002_int__le__induct,axiom,
    ! [Ia: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),K)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,Ia) ) ) ) ).

% int_le_induct
tff(fact_2003_int__less__induct,axiom,
    ! [Ia: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ia),K)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,Ia) ) ) ) ).

% int_less_induct
tff(fact_2004_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_2005_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_2006_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_2007_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_2008_ln__mult,axiom,
    ! [Xb: real,Y: 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)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2009_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2010_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int,K3: 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),K3),D2))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Z3)
             => ( aa(int,$o,P,X3)
              <=> aa(int,$o,P1,X3) ) )
         => ( ? [X_12: int] : aa(int,$o,P1,X_12)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_2011_plusinfinity,axiom,
    ! [D2: int,P2: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int,K3: 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),K3),D2))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X3)
             => ( aa(int,$o,P,X3)
              <=> aa(int,$o,P2,X3) ) )
         => ( ? [X_12: int] : aa(int,$o,P2,X_12)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_2012_int__induct,axiom,
    ! [P: fun(int,$o),K: int,Ia: int] :
      ( aa(int,$o,P,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,Ia) ) ) ) ).

% int_induct
tff(fact_2013_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_2014_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int] :
            ( aa(int,$o,P,X3)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X: int] :
              ( aa(int,$o,P,X)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_2015_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_2016_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_2017_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_2018_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_le_swap2
tff(fact_2019_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_2020_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% compl_mono
tff(fact_2021_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_2022_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_less_swap2
tff(fact_2023_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ~ ! [N: nat] : Xb != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_2024_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% diff_shunt_var
tff(fact_2025_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_2026_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(
            K = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
              aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
                $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_2027_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A4: real,B4: real,C4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A4),B4)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B4),C4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),B4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B4),C4)
                 => aa(real,$o,aa(real,fun(real,$o),P,A4),C4) ) ) ) )
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
               => ? [D3: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
                    & ! [A4: real,B4: real] :
                        ( ( aa(real,$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,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B4),A4)),D3) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A4),B4) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_2028_set__decode__0,axiom,
    ! [Xb: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xb))
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Xb) ) ).

% set_decode_0
tff(fact_2029_divmod__step__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_bb(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_2030_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(set(nat),nat,nat_set_encode,A3))
      <=> ~ member(nat,zero_zero(nat),A3) ) ) ).

% even_set_encode_iff
tff(fact_2031_Diff__idemp,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ).

% Diff_idemp
tff(fact_2032_Diff__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
    <=> ( member(A,C2,A3)
        & ~ member(A,C2,B3) ) ) ).

% Diff_iff
tff(fact_2033_DiffI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,A3)
     => ( ~ member(A,C2,B3)
       => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) ) ) ).

% DiffI
tff(fact_2034_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_2035_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),uminus_uminus(set(A)),B3) )
    <=> ( A3 = B3 ) ) ).

% Compl_eq_Compl_iff
tff(fact_2036_Compl__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3))
    <=> ~ member(A,C2,A3) ) ).

% Compl_iff
tff(fact_2037_ComplI,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( ~ member(A,C2,A3)
     => member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3)) ) ).

% ComplI
tff(fact_2038_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_2039_of__int__of__bool,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: $o] : aa(int,A,ring_1_of_int(A),aa($o,int,zero_neq_one_of_bool(int),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% of_int_of_bool
tff(fact_2040_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_2041_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_2042_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_2043_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_2044_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_2045_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_numeral
tff(fact_2046_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_2047_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_2048_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_2049_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_2050_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_2051_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_2052_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_2053_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_2054_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_2055_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_2056_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_2057_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_2058_set__encode__inverse,axiom,
    ! [A3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A3)) = A3 ) ) ).

% set_encode_inverse
tff(fact_2059_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_2060_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_2061_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_2062_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_2063_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_2064_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_2065_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_2066_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_2067_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_2068_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_2069_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_2070_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_2071_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_2072_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_2073_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xb)),Nb) )
        <=> ( Y = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_2074_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: nat,Y: int] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xb)),Nb) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_2075_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_2076_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_2077_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_2078_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_2079_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_2080_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_2081_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb) )
        <=> ( Y = 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_2082_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: nat,Y: 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),Y) )
        <=> ( aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2083_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_2084_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_2085_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_2086_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_2087_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aTP_Lamp_bc(set(A),fun(A,$o),A3)) ).

% Compl_eq
tff(fact_2088_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_bd(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% set_diff_eq
tff(fact_2089_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = collect(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),minus_minus(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3))) ).

% minus_set_def
tff(fact_2090_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : collect(A,aTP_Lamp_be(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_2091_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3))) ).

% uminus_set_def
tff(fact_2092_double__complement,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = A3 ).

% double_complement
tff(fact_2093_DiffD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
     => ~ member(A,C2,B3) ) ).

% DiffD2
tff(fact_2094_DiffD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
     => member(A,C2,A3) ) ).

% DiffD1
tff(fact_2095_ComplD,axiom,
    ! [A: $tType,C2: A,A3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A3))
     => ~ member(A,C2,A3) ) ).

% ComplD
tff(fact_2096_DiffE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))
     => ~ ( member(A,C2,A3)
         => member(A,C2,B3) ) ) ).

% DiffE
tff(fact_2097_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),Xb)) ) ).

% mult_of_int_commute
tff(fact_2098_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,Y: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),Xb),Y)) = 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),Y)) ) ).

% of_int_max
tff(fact_2099_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q2: product_prod(A,B),F2: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P3: product_prod(A,B)] :
      ( ! [X3: A,Y3: B] :
          ( ( aa(B,product_prod(A,B),product_Pair(A,B,X3),Y3) = Q2 )
         => ( aa(B,C,aa(A,fun(B,C),F2,X3),Y3) = aa(B,C,aa(A,fun(B,C),G,X3),Y3) ) )
     => ( ( P3 = Q2 )
       => ( aa(product_prod(A,B),C,product_case_prod(A,B,C,F2),P3) = aa(product_prod(A,B),C,product_case_prod(A,B,C,G),Q2) ) ) ) ).

% split_cong
tff(fact_2100_finite__set__decode,axiom,
    ! [Nb: nat] : aa(set(nat),$o,finite_finite2(nat),nat_set_decode(Nb)) ).

% finite_set_decode
tff(fact_2101_set__encode__eq,axiom,
    ! [A3: set(nat),B3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( aa(set(nat),$o,finite_finite2(nat),B3)
       => ( ( aa(set(nat),nat,nat_set_encode,A3) = aa(set(nat),nat,nat_set_encode,B3) )
        <=> ( A3 = B3 ) ) ) ) ).

% set_encode_eq
tff(fact_2102_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_2103_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_2104_set__encode__inf,axiom,
    ! [A3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),A3)
     => ( aa(set(nat),nat,nat_set_encode,A3) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_2105_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_2106_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_2107_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_neg_numeral
tff(fact_2108_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int,K3: int] :
            ( aa(int,$o,P,X3)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D2))) )
       => ( ? [X_13: int] : aa(int,$o,P,X_13)
        <=> ? [X2: int] :
              ( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P,X2) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2109_aset_I7_J,axiom,
    ! [D5: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
         => ( 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_2110_aset_I5_J,axiom,
    ! [D5: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,A3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( 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_2111_aset_I4_J,axiom,
    ! [D5: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,A3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( ( X != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_2112_aset_I3_J,axiom,
    ! [D5: int,Ta: int,A3: 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)),A3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( ( X = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_2113_bset_I7_J,axiom,
    ! [D5: int,Ta: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,B3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( 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_2114_bset_I5_J,axiom,
    ! [D5: int,B3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B3)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( 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_2115_bset_I4_J,axiom,
    ! [D5: int,Ta: int,B3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,Ta,B3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( X != Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_2116_bset_I3_J,axiom,
    ! [D5: int,Ta: int,B3: 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)),B3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( X = Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_2117_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_2118_aset_I8_J,axiom,
    ! [D5: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
         => ( 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_2119_aset_I6_J,axiom,
    ! [D5: int,Ta: int,A3: 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)),A3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa2) ) ) )
           => ( 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_2120_bset_I8_J,axiom,
    ! [D5: int,Ta: int,B3: 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)),B3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B3)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( 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_2121_bset_I6_J,axiom,
    ! [D5: int,B3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B3)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( 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_2122_cpmi,axiom,
    ! [D5: int,P: fun(int,$o),P2: fun(int,$o),B3: 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,$o,P,X3)
            <=> aa(int,$o,P2,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb3: int] :
                      ( member(int,Xb3,B3)
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
             => ( aa(int,$o,P,X3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5)) ) )
         => ( ! [X3: int,K3: 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),K3),D5))) )
           => ( ? [X_13: int] : aa(int,$o,P,X_13)
            <=> ( ? [X2: int] :
                    ( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P2,X2) )
                | ? [X2: int] :
                    ( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,B3)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X2)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2123_cppi,axiom,
    ! [D5: int,P: fun(int,$o),P2: fun(int,$o),A3: 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,$o,P,X3)
            <=> aa(int,$o,P2,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb3: int] :
                      ( member(int,Xb3,A3)
                     => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa) ) ) )
             => ( aa(int,$o,P,X3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)) ) )
         => ( ! [X3: int,K3: 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),K3),D5))) )
           => ( ? [X_13: int] : aa(int,$o,P,X_13)
            <=> ( ? [X2: int] :
                    ( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P2,X2) )
                | ? [X2: int] :
                    ( member(int,X2,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,A3)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa3),X2)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2124_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_bf(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_2125_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_2126_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,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int))))
          & ! [Y2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y2)),Xb)
                & aa(A,$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),Y2),one_one(int)))) )
             => ( Y2 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_2127_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_2128_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),product_Pair(nat,nat,zero_zero(nat)),Mb),
        aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_bg(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_2129_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_2130_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),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) = Y ) ) ) ) ).

% round_unique
tff(fact_2131_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: 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),Y),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)),Y)
           => 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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_2132_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_2133_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_2134_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_2135_tanh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xb)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% tanh_real_less_iff
tff(fact_2136_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_2137_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_2138_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_2139_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_2140_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_2141_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2142_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_2143_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_2144_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_2145_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_2146_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_2147_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))
        <=> ( ( Nb = zero_zero(nat) )
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_2148_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_2149_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_2150_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_2151_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_2152_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_2153_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_2154_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_2155_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_2156_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xb)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_2157_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_2158_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K: 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),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2159_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_2160_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_2161_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_2162_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_2163_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_2164_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_2165_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_2166_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_2167_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_2168_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_2169_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_2170_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_2171_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_2172_real__less__rsqrt,axiom,
    ! [Xb: real,Y: 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)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_2173_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% take_bit_int_less_exp
tff(fact_2174_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) )
        <=> 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_2175_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_2176_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_2177_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2178_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_2179_real__less__lsqrt,axiom,
    ! [Xb: real,Y: 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)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_2180_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_2181_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2182_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2183_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_12: nat] : aa(nat,$o,P,X_12)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P,N)
            & aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_2184_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_2185_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_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K)
     => ( 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),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_2193_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_2194_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(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_2195_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) = 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)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2196_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_bh(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_2197_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_bi(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_2198_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_bj(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_2199_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)) )
     => ~ 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_2200_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),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_2201_set__encode__insert,axiom,
    ! [A3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),A3)
     => ( ~ member(nat,Nb,A3)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Nb),A3)) = 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,A3)) ) ) ) ).

% set_encode_insert
tff(fact_2202_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_bk(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_2203_insert__absorb2,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3) ).

% insert_absorb2
tff(fact_2204_insert__iff,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),A3))
    <=> ( ( A2 = B2 )
        | member(A,A2,A3) ) ) ).

% insert_iff
tff(fact_2205_insertCI,axiom,
    ! [A: $tType,A2: A,B3: set(A),B2: A] :
      ( ( ~ member(A,A2,B3)
       => ( A2 = B2 ) )
     => member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3)) ) ).

% insertCI
tff(fact_2206_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( bit1(X32) = bit1(Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_2207_singletonI,axiom,
    ! [A: $tType,A2: A] : member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ).

% singletonI
tff(fact_2208_finite__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3))
    <=> aa(set(A),$o,finite_finite2(A),A3) ) ).

% finite_insert
tff(fact_2209_insert__subset,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),B3)
    <=> ( member(A,Xb,B3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ) ).

% insert_subset
tff(fact_2210_insert__Diff1,axiom,
    ! [A: $tType,Xb: A,B3: set(A),A3: set(A)] :
      ( member(A,Xb,B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ) ) ).

% insert_Diff1
tff(fact_2211_Diff__insert0,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B3: set(A)] :
      ( ~ member(A,Xb,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) ) ) ).

% Diff_insert0
tff(fact_2212_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_2213_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_bl(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2214_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : collect(A,aa(A,fun(A,$o),fequal(A),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_2215_singleton__conv,axiom,
    ! [A: $tType,A2: A] : collect(A,aTP_Lamp_bm(A,fun(A,$o),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_2216_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_2217_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),F4) = zero_zero(B) )
          <=> ! [X2: A] :
                ( member(A,X2,F4)
               => ( aa(A,B,F2,X2) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_2218_sum_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_2219_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2220_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2221_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),bot_bot(set(A))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_2222_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : set_or1337092689740270186AtMost(A,A2,A2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_2223_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) ).

% insert_Diff_single
tff(fact_2224_finite__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)))
    <=> aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) ) ).

% finite_Diff_insert
tff(fact_2225_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_2226_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_2227_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_bn(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_2228_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_2229_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_2230_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
    <=> ~ member(A,B2,A3) ) ).

% subset_Compl_singleton
tff(fact_2231_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_bp(fun(B,int),fun(B,A),F2)),A3) ) ).

% of_int_sum
tff(fact_2232_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_2233_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_2234_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num] : unique8689654367752047608divmod(A,Mb,one2) = aa(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_2235_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)),insert2(A),Xb),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_2236_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ).

% sum.cl_ivl_Suc
tff(fact_2237_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_2238_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V2: num] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),aa(num,nat,numeral_numeral(nat),V2)) = 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),V2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2239_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,bit0(Nb)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_2240_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_2241_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V2: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),aa(num,nat,numeral_numeral(nat),V2)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Mb),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2242_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,bit1(Nb)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_2243_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bq(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A3)
            & member(nat,zero_zero(nat),A3) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_2244_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D2: fun(nat,A),A3: 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_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A3)
            & member(nat,zero_zero(nat),A3) ),
            divide_divide(A,aa(nat,A,C2,zero_zero(nat)),aa(nat,A,D2,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_2245_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),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_2246_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),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_2247_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),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_bs(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(6)
tff(fact_2248_insert__compr,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_bt(A,fun(set(A),fun(A,$o)),A2),B3)) ).

% insert_compr
tff(fact_2249_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),collect(A,P)) = collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bu(A,fun(fun(A,$o),fun(A,$o)),A2),P)) ).

% insert_Collect
tff(fact_2250_mk__disjoint__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( member(A,A2,A3)
     => ? [B9: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B9) )
          & ~ member(A,A2,B9) ) ) ).

% mk_disjoint_insert
tff(fact_2251_insert__commute,axiom,
    ! [A: $tType,Xb: A,Y: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) ).

% insert_commute
tff(fact_2252_insert__eq__iff,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A,B3: set(A)] :
      ( ~ member(A,A2,A3)
     => ( ~ member(A,B2,B3)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3) )
        <=> $ite(
              A2 = B2,
              A3 = B3,
              ? [C7: set(A)] :
                ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),C7) )
                & ~ member(A,B2,C7)
                & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),C7) )
                & ~ member(A,A2,C7) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_2253_insert__absorb,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( member(A,A2,A3)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) = A3 ) ) ).

% insert_absorb
tff(fact_2254_insert__ident,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B3: set(A)] :
      ( ~ member(A,Xb,A3)
     => ( ~ member(A,Xb,B3)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),B3) )
        <=> ( A3 = B3 ) ) ) ) ).

% insert_ident
tff(fact_2255_Set_Oset__insert,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] :
      ( member(A,Xb,A3)
     => ~ ! [B9: set(A)] :
            ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),B9) )
           => member(A,Xb,B9) ) ) ).

% Set.set_insert
tff(fact_2256_insertI2,axiom,
    ! [A: $tType,A2: A,B3: set(A),B2: A] :
      ( member(A,A2,B3)
     => member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3)) ) ).

% insertI2
tff(fact_2257_insertI1,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) ).

% insertI1
tff(fact_2258_insertE,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),A3))
     => ( ( A2 != B2 )
       => member(A,A2,A3) ) ) ).

% insertE
tff(fact_2259_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(member(A,Xb,A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3))) ) ) ) ).

% sum.insert_if
tff(fact_2260_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A3)
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_2261_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => ( aa(A,B,G,X3) = zero_zero(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_2262_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : bit0(X22) != bit1(X32) ).

% verit_eq_simplify(14)
tff(fact_2263_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != bit1(X32) ).

% verit_eq_simplify(12)
tff(fact_2264_singletonD,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( member(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_2265_singleton__iff,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( member(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_2266_doubleton__eq__iff,axiom,
    ! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),D2),bot_bot(set(A)))) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_2267_insert__not__empty,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_2268_singleton__inject,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_2269_finite_OinsertI,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) ) ).

% finite.insertI
tff(fact_2270_insert__mono,axiom,
    ! [A: $tType,C5: set(A),D5: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),D5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),C5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),D5)) ) ).

% insert_mono
tff(fact_2271_subset__insert,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B3: set(A)] :
      ( ~ member(A,Xb,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),B3))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ) ).

% subset_insert
tff(fact_2272_subset__insertI,axiom,
    ! [A: $tType,B3: set(A),A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) ).

% subset_insertI
tff(fact_2273_subset__insertI2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3)) ) ).

% subset_insertI2
tff(fact_2274_insert__subsetI,axiom,
    ! [A: $tType,Xb: A,A3: set(A),X5: set(A)] :
      ( member(A,Xb,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),X5)),A3) ) ) ).

% insert_subsetI
tff(fact_2275_insert__Diff__if,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),B3) = $ite(member(A,Xb,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ).

% insert_Diff_if
tff(fact_2276_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K5: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,K5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),K5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),K5)) ) ) ).

% sum_mono
tff(fact_2277_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_bv(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),A3)) ) ).

% sum.distrib
tff(fact_2278_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B3: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(B),$o,finite_finite2(B),B3)
           => ( 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_bx(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B3),G),R)),A3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_bz(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R)),B3) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2279_sum__diff1__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),A2: A] :
      aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(A,nat,F2,A2)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)) ).

% sum_diff1_nat
tff(fact_2280_Collect__conv__if2,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ca(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_2281_Collect__conv__if,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cb(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_2282_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_2283_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_2284_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),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),F2),A3)),aa(A,B,F2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ) ).

% sum_diff1
tff(fact_2285_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ).

% sum_nonneg
tff(fact_2286_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_2287_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(B,A),I5: set(B),G: fun(B,A),Ia: B] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),I5) )
         => ( ! [I2: B] :
                ( member(B,I2,I5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,I2)),aa(B,A,G,I2)) )
           => ( member(B,Ia,I5)
             => ( aa(set(B),$o,finite_finite2(B),I5)
               => ( aa(B,A,F2,Ia) = aa(B,A,G,Ia) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_2288_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A3)
         => ( ! [X3: nat] :
                ( member(nat,aa(nat,nat,suc,X3),A3)
               => ( aa(nat,A,F2,aa(nat,nat,suc,X3)) = aa(nat,A,G,aa(nat,nat,suc,X3)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_2289_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$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_cc(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),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),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_2290_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X23: num] : Y != bit0(X23)
       => ~ ! [X33: num] : Y != bit1(X33) ) ) ).

% num.exhaust
tff(fact_2291_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)),insert2(A),A4),A6)
             => ~ aa(set(A),$o,finite_finite2(A),A6) ) ) ) ).

% finite.cases
tff(fact_2292_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)),insert2(A),A7),A8) )
            & aa(set(A),$o,finite_finite2(A),A8) ) ) ) ).

% finite.simps
tff(fact_2293_finite__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,P,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,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F5)) ) ) )
         => aa(set(A),$o,P,F4) ) ) ) ).

% finite_induct
tff(fact_2294_finite__ne__induct,axiom,
    ! [A: $tType,F4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ( F4 != bot_bot(set(A)) )
       => ( ! [X3: A] : aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))
         => ( ! [X3: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ member(A,X3,F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F5)) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_2295_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),A3: set(A)] :
      ( ! [A6: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A6)
         => aa(set(A),$o,P,A6) )
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X3: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ member(A,X3,F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F5)) ) ) )
         => aa(set(A),$o,P,A3) ) ) ) ).

% infinite_finite_induct
tff(fact_2296_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)),insert2(A),A2),bot_bot(set(A))))
    <=> ( ( X5 = bot_bot(set(A)) )
        | ( X5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_2297_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_2298_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_2299_Diff__insert__absorb,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] :
      ( ~ member(A,Xb,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2300_Diff__insert2,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))),B3) ).

% Diff_insert2
tff(fact_2301_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( member(A,A2,A3)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = A3 ) ) ).

% insert_Diff
tff(fact_2302_Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_2303_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),Xb: A,C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),C5)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C5))
        & ~ member(A,Xb,A3) ) ) ).

% subset_Diff_insert
tff(fact_2304_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_cd(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_2305_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [Ia: A,A3: set(A),F2: fun(A,B)] :
          ( member(A,Ia,A3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ia),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
           => ( aa(set(A),$o,finite_finite2(A),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ia)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ) ) ).

% member_le_sum
tff(fact_2306_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X3: A] :
          ( member(A,X3,A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X3)),aa(A,nat,F2,X3)) )
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_ce(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_2307_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cf(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2308_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_cg(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2309_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S: set(A),Ta: set(B),G: fun(B,C),Ia: fun(B,A),F2: fun(A,C)] :
          ( aa(set(A),$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,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X3)) )
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ? [Xa: B] :
                        ( member(B,Xa,Ta)
                        & ( aa(B,A,Ia,Xa) = X3 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X3)),aa(B,C,G,Xa)) ) )
               => aa(C,$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),F2),S)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_2310_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3) = zero_zero(B) )
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => ( aa(A,B,F2,X2) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_2311_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
           => ( ? [X: A] :
                  ( member(A,X,A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,G,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),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_2312_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X1: A,Y1: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S2)
             => ( ! [X3: B] :
                    ( member(B,X3,S2)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X3)),aa(B,A,G,X3)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2)) ) ) ) ) ) ).

% sum.related
tff(fact_2313_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A] :
                  ( member(A,X3,A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)) ) ) ) ) ).

% sum_strict_mono
tff(fact_2314_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T4: set(B),S2: set(A),Ia: fun(B,A),J: fun(A,B),T3: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S4)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S4))
                 => ( aa(B,A,Ia,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),S4))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T4)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T4))
                     => ( aa(A,B,J,aa(B,A,Ia,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T4))
                       => member(A,aa(B,A,Ia,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S4)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S4)
                         => ( aa(A,C,G,A4) = zero_zero(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T4)
                           => ( 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,G,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),G),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_2315_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X2: A] :
            ( member(A,X2,A3)
            & ( aa(A,nat,F2,X2) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( ( X2 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_2316_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),Nb: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = aa(nat,nat,suc,Nb) )
     => ? [X3: A] :
          ( member(A,X3,A3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X3)) ) ) ).

% sum_SucD
tff(fact_2317_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3) = one_one(nat) )
      <=> ? [X2: A] :
            ( member(A,X2,A3)
            & ( aa(A,nat,F2,X2) = one_one(nat) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( ( X2 != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2318_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_2319_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),P: fun(set(A),$o),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X3: A,S5: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S5)
                 => ( ! [Y2: A] :
                        ( member(A,Y2,S5)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y2)),aa(A,B,F2,X3)) )
                   => ( aa(set(A),$o,P,S5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),S5)) ) ) )
             => aa(set(A),$o,P,S2) ) ) ) ) ).

% finite_ranking_induct
tff(fact_2320_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,P,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,P,A6)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),A6)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_2321_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,P,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,P,A6)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),A6)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_2322_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa(num,nat,numeral_numeral(nat),bit1(Nb)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(Nb))) ).

% eval_nat_numeral(3)
tff(fact_2323_finite__subset__induct_H,axiom,
    ! [A: $tType,F4: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( member(A,A4,A3)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A3)
                   => ( ~ member(A,A4,F5)
                     => ( aa(set(A),$o,P,F5)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),F5)) ) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_2324_finite__subset__induct,axiom,
    ! [A: $tType,F4: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( member(A,A4,A3)
                 => ( ~ member(A,A4,F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),F5)) ) ) ) )
           => aa(set(A),$o,P,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_2325_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),B3: B,Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S) = B3 )
             => ( member(A,Ia,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ia)),B3) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2326_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S) = zero_zero(B) )
             => ( member(A,Ia,S)
               => ( aa(A,B,F2,Ia) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2327_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)),insert2(A),A2),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_2328_infinite__coinduct,axiom,
    ! [A: $tType,X5: fun(set(A),$o),A3: set(A)] :
      ( aa(set(A),$o,X5,A3)
     => ( ! [A6: set(A)] :
            ( aa(set(A),$o,X5,A6)
           => ? [X: A] :
                ( member(A,X,A6)
                & ( aa(set(A),$o,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
                  | ~ aa(set(A),$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)),insert2(A),X),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% infinite_coinduct
tff(fact_2329_finite__empty__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,P,A3)
       => ( ! [A4: A,A6: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A6)
             => ( member(A,A4,A6)
               => ( aa(set(A),$o,P,A6)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_2330_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),B3)) ) ).

% Diff_single_insert
tff(fact_2331_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),B3))
    <=> $ite(member(A,Xb,A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))),B3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)) ) ).

% subset_insert_iff
tff(fact_2332_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)),insert2(nat),aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_2333_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)),insert2(nat),Mb),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2334_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)),insert2(nat),aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2335_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)),insert2(nat),Mb),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) = set_or1337092689740270186AtMost(nat,Mb,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_2336_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),collect(A,aTP_Lamp_ch(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2337_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),Ia: nat,Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,Ia,Xb))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_2338_Compl__insert,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_2339_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,I5: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ci(A,fun(nat,fun(nat,A)),Xb),Mb)),I5) = 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)),I5)) ) ).

% sum_power_add
tff(fact_2340_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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_cj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2341_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_2342_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),Ia: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( member(A,Ia,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,Ia))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_2343_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) ) ) ) ) ).

% sum_pos
tff(fact_2344_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G,X3) = zero_zero(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),S2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_2345_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,H,X3) = zero_zero(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_2346_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S2) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_2347_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T3) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_2348_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3))
                   => ( aa(A,B,G,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),B3))
                     => ( aa(A,B,H,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),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),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B3) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_2349_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3))
                   => ( aa(A,B,G,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),B3))
                     => ( aa(A,B,H,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B3) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C5) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_2350_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B3: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ).

% sum.subset_diff
tff(fact_2351_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = 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),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3)) ) ) ) ) ).

% sum_diff
tff(fact_2352_sum__diff__nat,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B3)) ) ) ) ).

% sum_diff_nat
tff(fact_2353_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_2354_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,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),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2355_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2356_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2357_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_2358_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_2359_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_2360_finite__remove__induct,axiom,
    ! [A: $tType,B3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,P,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),$o),ord_less_eq(set(A)),A6),B3)
                 => ( ! [X: A] :
                        ( member(A,X,A6)
                       => aa(set(A),$o,P,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)),insert2(A),X),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A6) ) ) ) )
         => aa(set(A),$o,P,B3) ) ) ) ).

% finite_remove_induct
tff(fact_2361_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B3: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ aa(set(A),$o,finite_finite2(A),B3)
         => aa(set(A),$o,P,B3) )
       => ( ! [A6: set(A)] :
              ( aa(set(A),$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),B3)
                 => ( ! [X: A] :
                        ( member(A,X,A6)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A6) ) ) ) )
         => aa(set(A),$o,P,B3) ) ) ) ).

% remove_induct
tff(fact_2362_finite__induct__select,axiom,
    ! [A: $tType,S2: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T5: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T5),S2)
             => ( aa(set(A),$o,P,T5)
               => ? [X: A] :
                    ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T5))
                    & aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),T5)) ) ) )
         => aa(set(A),$o,P,S2) ) ) ) ).

% finite_induct_select
tff(fact_2363_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),B3))
    <=> $ite(
          member(A,Xb,B3),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3),
          $ite(member(A,Xb,A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))),B3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)) ) ) ).

% 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)),insert2(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)),insert2(A),Xb),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_2366_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cf(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2367_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$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_ck(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,aa(nat,nat,suc,Nb))),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_Suc_diff
tff(fact_2368_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,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),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3)) ) ) ) ) ).

% sum_mono2
tff(fact_2369_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_2370_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_2371_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2372_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_2373_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_2374_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_2375_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_2376_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_2377_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B3: set(A),A3: set(A),B2: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => ( member(A,B2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,B2))
               => ( ! [X3: A] :
                      ( member(A,X3,B3)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_2378_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_2379_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: 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_cl(fun(nat,A),fun(nat,A),F2)),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,F2,Mb)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_2380_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$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_cm(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_telescope''
tff(fact_2381_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)),insert2(nat),Nb),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_2382_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb))) ) ).

% mask_eq_sum_exp
tff(fact_2383_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_2384_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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_cn(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.in_pairs
tff(fact_2385_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_2386_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)))),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_2387_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bk(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_2388_arith__series__nat,axiom,
    ! [A2: nat,D2: 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_co(nat,fun(nat,fun(nat,nat)),A2),D2)),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),D2))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_2389_dbl__dec__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit1(one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_2390_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_2391_dbl__inc__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit1(one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_2392_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_2393_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_2394_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_2395_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_2396_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_2397_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V2: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Mb) = aa(num,int,numeral_numeral(int),V2) )
    <=> ( Mb = aa(num,nat,numeral_numeral(nat),V2) ) ) ).

% int_eq_iff_numeral
tff(fact_2398_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_2399_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_2400_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_2401_int__dvd__int__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),Mb),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> dvd_dvd(nat,Mb,Nb) ) ).

% int_dvd_int_iff
tff(fact_2402_dbl__dec__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,one_one(A)) = one_one(A) ) ) ).

% dbl_dec_simps(3)
tff(fact_2403_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_2404_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_2405_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_2406_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_2407_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_2408_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ).

% of_nat_numeral
tff(fact_2409_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_2410_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_2411_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_2412_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_2413_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_2414_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_2415_dbl__inc__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,zero_zero(A)) = one_one(A) ) ) ).

% dbl_inc_simps(2)
tff(fact_2416_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o] : aa(nat,A,semiring_1_of_nat(A),aa($o,nat,zero_neq_one_of_bool(nat),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% of_nat_of_bool
tff(fact_2417_dbl__inc__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% dbl_inc_simps(4)
tff(fact_2418_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bit1(K)) ) ).

% dbl_inc_simps(5)
tff(fact_2419_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cp(fun(B,nat),fun(B,A),F2)),A3) ) ).

% of_nat_sum
tff(fact_2420_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_2421_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_2422_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_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_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_2425_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_2426_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_dec_simps(1)
tff(fact_2427_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_inc_simps(1)
tff(fact_2428_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_2429_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_2430_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_2431_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_2432_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_2433_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_2434_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,Ia: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Ia)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Ia)),Nb)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2435_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ia: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Ia)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Ia)),Nb)),Xb) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2436_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,Ia: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Ia)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Ia)),Nb)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2437_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ia: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Ia)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Ia)),Nb)),Xb) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2438_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_2439_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_2440_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xb)) ) ).

% mult_of_nat_commute
tff(fact_2441_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% int_cases2
tff(fact_2442_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M2: nat,N: nat] : Z != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_diff_cases
tff(fact_2443_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_2444_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_2445_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_2446_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_2447_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_2448_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_2449_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Ia: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Ia)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_2450_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_2451_int__ops_I3_J,axiom,
    ! [Nb: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,int,numeral_numeral(int),Nb) ).

% int_ops(3)
tff(fact_2452_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_2453_int__of__nat__induct,axiom,
    ! [P: fun(int,$o),Z: int] :
      ( ! [N: nat] : aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),N))
     => ( ! [N: nat] : aa(int,$o,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))
       => aa(int,$o,P,Z) ) ) ).

% int_of_nat_induct
tff(fact_2454_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% int_cases
tff(fact_2455_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_2456_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_2457_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ~ ! [N: nat] : K != aa(nat,int,semiring_1_of_nat(int),N) ) ).

% nonneg_int_cases
tff(fact_2458_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ? [N: nat] : K = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% zero_le_imp_eq_int
tff(fact_2459_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_2460_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_2461_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_2462_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_2463_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_2464_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z)
    <=> ? [N2: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% zle_iff_zadd
tff(fact_2465_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_2466_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xb),Y)) = 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),Y)) ) ).

% of_nat_max
tff(fact_2467_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_2468_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_2469_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),Xb)) ) ) ).

% ex_less_of_nat_mult
tff(fact_2470_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_2471_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ! [Y2: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xb)) ) ).

% reals_Archimedean3
tff(fact_2472_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N: nat] : Mb != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( Mb != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ) ) ).

% int_cases4
tff(fact_2473_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_2474_int__ops_I4_J,axiom,
    ! [A2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),one_one(int)) ).

% int_ops(4)
tff(fact_2475_int__Suc,axiom,
    ! [Nb: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) ).

% int_Suc
tff(fact_2476_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
    <=> ? [N2: 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,N2))) ) ).

% zless_iff_Suc_zadd
tff(fact_2477_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ~ ! [N: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% nonpos_int_cases
tff(fact_2478_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_2479_int__sum,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: 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),F2),A3)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7311177749621191930dd_sum(A,int),aTP_Lamp_cq(fun(A,nat),fun(A,int),F2)),A3) ).

% int_sum
tff(fact_2480_sum__nth__roots,axiom,
    ! [Nb: nat,C2: 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_cr(complex,complex)),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_av(nat,fun(complex,fun(complex,$o)),Nb),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_2481_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_cr(complex,complex)),collect(complex,aTP_Lamp_cs(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_2482_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_2483_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2484_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ~ ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% pos_int_cases
tff(fact_2485_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) )
       => ~ ! [N: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% int_cases3
tff(fact_2486_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_2487_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_2488_zmult__zless__mono2__lemma,axiom,
    ! [Ia: int,J: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ia),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),Ia)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_2489_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_2490_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_2491_negD,axiom,
    ! [Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),zero_zero(int))
     => ? [N: nat] : Xb = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% negD
tff(fact_2492_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl_inc(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_2493_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_2494_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)),insert2(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_2495_simp__from__to,axiom,
    ! [Ia: int,J: int] :
      set_or1337092689740270186AtMost(int,Ia,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),Ia),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J))) ).

% simp_from_to
tff(fact_2496_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
         => ~ ! [N: nat] : ~ aa(A,$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,N)))),E2) ) ) ).

% nat_approx_posE
tff(fact_2497_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_2498_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_2499_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C2: 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)),C2)
       => ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
             => 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),M2)),Xb)),C2) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2500_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ~ ! [N: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% neg_int_cases
tff(fact_2501_zdiff__int__split,axiom,
    ! [P: fun(int,$o),Xb: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xb)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Xb)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_2502_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_2503_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_2504_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_2505_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D2: 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_ct(A,fun(A,fun(nat,A)),A2),D2)),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)),D2))) ) ).

% double_arith_series
tff(fact_2506_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D2: 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_cu(A,fun(A,fun(nat,A)),A2),D2)),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)),D2))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_2507_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,product_case_prod(nat,nat,A,aTP_Lamp_cv(nat,fun(nat,A))),divmod_nat(Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_2508_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_2509_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_cw(real,fun(nat,real),Xb)) ) ) ) ).

% ln_series
tff(fact_2510_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_cy(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_2511_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_2512_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_cz(A,A),Nb,zero_zero(A)) ) ).

% of_nat_code
tff(fact_2513_lessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),Xb) = aa(A,set(A),set_ord_lessThan(A),Y) )
        <=> ( Xb = Y ) ) ) ).

% lessThan_eq_iff
tff(fact_2514_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,K: A] :
          ( member(A,Ia,aa(A,set(A),set_ord_lessThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),K) ) ) ).

% lessThan_iff
tff(fact_2515_finite__lessThan,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% finite_lessThan
tff(fact_2516_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),Xb)),aa(A,set(A),set_ord_lessThan(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% lessThan_subset_iff
tff(fact_2517_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_2518_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_2519_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_2520_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_2521_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,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),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% sum.lessThan_Suc
tff(fact_2522_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_da(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2523_nat__int__comparison_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A2) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_2524_int__if,axiom,
    ! [P: $o,A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),
        $ite((P),A2,B2)) = $ite((P),aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_if
tff(fact_2525_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_2526_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_2527_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_2528_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = collect(A,aTP_Lamp_db(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_2529_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_2530_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_2531_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_2532_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_2533_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_2534_lessThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),K),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% lessThan_Suc
tff(fact_2535_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_2536_finite__nat__iff__bounded,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ? [K2: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_lessThan(nat),K2)) ) ).

% finite_nat_iff_bounded
tff(fact_2537_finite__nat__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_bounded
tff(fact_2538_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_2539_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Nb: nat,Ia: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Nb),Ia) = semiri8178284476397505188at_aux(A,Inc,Nb,aa(A,A,Inc,Ia)) ) ).

% of_nat_aux.simps(2)
tff(fact_2540_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Ia: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),Ia) = Ia ) ).

% of_nat_aux.simps(1)
tff(fact_2541_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_2542_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: 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_dc(fun(nat,A),fun(nat,fun(nat,A)),G),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),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.nat_diff_reindex
tff(fact_2543_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Nb: A] :
          ( ! [X3: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X3)),aa(A,nat,P,X3))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),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_dd(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),Nb)) ) ) ) ).

% sum_diff_distrib
tff(fact_2544_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),K6) )
        <=> ? [N6: nat] :
            ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% lemma_NBseq_def2
tff(fact_2545_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),K6) )
        <=> ? [N6: nat] :
            ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% lemma_NBseq_def
tff(fact_2546_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_2547_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_2548_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_2549_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
        <=> ? [K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Nb)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K2)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2550_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Nb: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2551_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2552_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2553_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_2554_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cf(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2555_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,Mb)) ) ).

% sum_lessThan_telescope'
tff(fact_2556_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ck(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Mb)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2557_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cf(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2558_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_2559_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_2560_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_2561_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_2562_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_2563_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_df(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_dg(A,fun(A,fun(nat,fun(nat,A))),Z),H),Mb)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) ) ).

% lemma_termdiff1
tff(fact_2564_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat,Y: 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,Y),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),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_dh(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% power_diff_sumr2
tff(fact_2565_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat,Y: 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,Y),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),Y)),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_di(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb)))) ) ).

% diff_power_eq_sum
tff(fact_2566_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2567_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F2: fun(nat,A),K5: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,P4)),K5) )
         => ( aa(A,$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),F2),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),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_2568_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_dj(A,fun(nat,fun(nat,A)),Xb),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% one_diff_power_eq'
tff(fact_2569_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_2570_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K)) ) ).

% pochhammer_minus'
tff(fact_2571_sum__split__even__odd,axiom,
    ! [F2: fun(nat,real),G: 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_dk(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G)),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_dl(fun(nat,real),fun(nat,real),F2)),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_dm(fun(nat,real),fun(nat,real),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ).

% sum_split_even_odd
tff(fact_2572_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_2573_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_2574_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => ( suminf(A,power_power(A,C2)) = divide_divide(A,one_one(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_2575_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2576_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_2577_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_dn(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_2578_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_2579_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xb)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ).

% norm_uminus_minus
tff(fact_2580_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_2581_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_2582_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,R2: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$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),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S)) ) ) ) ).

% norm_mult_less
tff(fact_2583_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E2) ) ) ).

% norm_triangle_lt
tff(fact_2584_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,R2: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$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),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S)) ) ) ) ).

% norm_add_less
tff(fact_2585_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C2)
         => aa(real,$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)),C2)) ) ) ).

% norm_add_leD
tff(fact_2586_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E2) ) ) ).

% norm_triangle_le
tff(fact_2587_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : aa(real,$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),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_2588_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_2589_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z))),E22)
           => aa(real,$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_2590_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_2591_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N3)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N3)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,F2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),N3) ) ) ) ) ).

% suminf_finite
tff(fact_2592_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_2593_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(real,$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),C2),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2594_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_do(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_2595_pi__series,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_dp(nat,real)) ).

% pi_series
tff(fact_2596_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
            & ~ 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),
              ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
              & ~ 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,K,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_2597_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Y )
     => ( Y = $ite(
            ( member(int,Xb,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xb)
                & ~ 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),
                  ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xb)
                  & ~ 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_2598_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_dq(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_2599_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_dr(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_2600_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_2601_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_2602_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_2603_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_2604_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ds(fun(B,nat),fun(B,A),F2)),A3) ) ).

% of_nat_prod
tff(fact_2605_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_dt(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_2606_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F2),A3)) = 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),F2)),A3) ) ).

% of_int_prod
tff(fact_2607_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3) = zero_zero(B) )
          <=> ? [X2: A] :
                ( member(A,X2,A3)
                & ( aa(A,B,F2,X2) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_2608_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_2609_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_2610_dvd__prod__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),A2: A,B2: B,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => ( ( B2 = aa(A,B,F2,A2) )
             => dvd_dvd(B,B2,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_2611_dvd__prodI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),A2: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => dvd_dvd(B,aa(A,B,F2,A2),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ) ).

% dvd_prodI
tff(fact_2612_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_2613_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_2614_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_2615_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).

% prod.insert
tff(fact_2616_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,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),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% prod.lessThan_Suc
tff(fact_2617_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_2618_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_2619_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xb: A] :
          ( sums(A,aTP_Lamp_dx(fun(nat,A),fun(nat,A),A2),Xb)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xb ) ) ) ).

% powser_sums_zero_iff
tff(fact_2620_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ).

% prod.cl_ivl_Suc
tff(fact_2621_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: 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(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(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),Y)))) ) ).

% and_numerals(7)
tff(fact_2622_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(nat,A,F2,N) = zero_zero(A)
         => sums(A,F2,zero_zero(A)) ) ) ).

% sums_0
tff(fact_2623_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S: A,Ta: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( sums(A,F2,S)
           => ( sums(A,G,Ta)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S),Ta) ) ) ) ) ).

% sums_le
tff(fact_2624_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Ia: nat,F2: fun(nat,A)] : sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dy(nat,fun(fun(nat,A),fun(nat,A)),Ia),F2),aa(nat,A,F2,Ia)) ) ).

% sums_single
tff(fact_2625_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( sums(A,F2,A2)
         => ( sums(A,G,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% sums_add
tff(fact_2626_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),B3: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(B),$o,finite_finite2(B),B3)
           => ( 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_ea(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B3),G),R)),A3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_ec(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R)),B3) ) ) ) ) ).

% prod.swap_restrict
tff(fact_2627_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ).

% prod_mono
tff(fact_2628_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_nonneg
tff(fact_2629_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_pos
tff(fact_2630_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_ge_1
tff(fact_2631_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ? [X: A] :
                ( member(A,X,A3)
                & ( aa(A,B,F2,X) = zero_zero(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_2632_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ed(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_2633_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ee(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))
          <=> sums(A,F2,D2) ) ) ) ).

% sums_mult_iff
tff(fact_2634_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ef(A,fun(fun(nat,A),fun(nat,A)),C2),F2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),C2))
          <=> sums(A,F2,D2) ) ) ) ).

% sums_mult2_iff
tff(fact_2635_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_2636_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_2637_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_eg(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% prod.inter_filter
tff(fact_2638_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_2639_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ei(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_2640_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_2641_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X1: A,Y1: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S2)
             => ( ! [X3: B] :
                    ( member(B,X3,S2)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X3)),aa(B,A,G,X3)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2)) ) ) ) ) ) ).

% prod.related
tff(fact_2642_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(member(A,Xb,A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3))) ) ) ) ).

% prod.insert_if
tff(fact_2643_prod__dvd__prod__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => dvd_dvd(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B3)) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_2644_prod__dvd__prod__subset2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => ( ! [A4: A] :
                  ( member(A,A4,A3)
                 => dvd_dvd(B,aa(A,B,F2,A4),aa(A,B,G,A4)) )
             => dvd_dvd(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_2645_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T4: set(B),S2: set(A),Ia: fun(B,A),J: fun(A,B),T3: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S4)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S4))
                 => ( aa(B,A,Ia,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),S4))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T4)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T4))
                     => ( aa(A,B,J,aa(B,A,Ia,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T4))
                       => member(A,aa(B,A,Ia,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S4)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S4)
                         => ( aa(A,C,G,A4) = one_one(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T4)
                           => ( 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,G,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_2646_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ej(A,fun(fun(nat,A),fun(nat,A)),C2),F2),A2)
         => ( ( C2 != zero_zero(A) )
           => sums(A,F2,divide_divide(A,A2,C2)) ) ) ) ).

% sums_mult_D
tff(fact_2647_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),F2),S)
           => sums(A,F2,S) ) ) ) ).

% sums_Suc_imp
tff(fact_2648_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),F2),S)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_2649_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_el(fun(nat,A),fun(nat,A),F2),L)
         => sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_2650_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),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),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_2651_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_2652_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_2653_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F2: fun(nat,A),S: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
             => ( aa(nat,A,F2,I2) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_em(nat,fun(fun(nat,A),fun(nat,A)),Nb),F2),S)
          <=> sums(A,F2,S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_2654_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N3)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N3)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => sums(A,F2,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),N3)) ) ) ) ).

% sums_finite
tff(fact_2655_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),collect(nat,P))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_en(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),collect(nat,P))) ) ) ).

% sums_If_finite
tff(fact_2656_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A3)
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eo(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A3)) ) ) ).

% sums_If_finite_set
tff(fact_2657_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),collect(A,aTP_Lamp_ep(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_2658_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: 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_eq(fun(nat,A),fun(nat,fun(nat,A)),G),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),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.nat_diff_reindex
tff(fact_2659_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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_er(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% prod.atLeastAtMost_rev
tff(fact_2660_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Ia: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( member(A,Ia,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,Ia))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_2661_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I5)) ) ) ) ) ).

% less_1_prod
tff(fact_2662_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B3: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ).

% prod.subset_diff
tff(fact_2663_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G,X3) = one_one(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_2664_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,H,X3) = one_one(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_2665_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S2) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_2666_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_2667_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3))
                   => ( aa(A,B,G,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),B3))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),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),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B3) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_2668_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A3: set(A),B3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3))
                   => ( aa(A,B,G,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),B3))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B3) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_2669_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_es(nat,fun(A,fun(nat,A)),Mb),Z),aa(nat,A,power_power(A,Z),Mb)) ) ).

% powser_sums_if
tff(fact_2670_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,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),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_2671_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_dx(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_2672_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_2673_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_2674_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb),S)
        <=> sums(A,F2,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),F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))) ) ) ).

% sums_iff_shift
tff(fact_2675_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A,Nb: nat] :
          ( sums(A,F2,S)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,fun(nat,A)),F2),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),F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_2676_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,fun(nat,A)),F2),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),F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb))))
        <=> sums(A,F2,S) ) ) ).

% sums_iff_shift'
tff(fact_2677_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S2: A,A3: set(nat),S4: A,F2: fun(nat,A)] :
          ( sums(A,G,S2)
         => ( aa(set(nat),$o,finite_finite2(nat),A3)
           => ( ( S4 = 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_eu(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ev(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2),S4) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_2678_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_2679_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% prod.lessThan_Suc_shift
tff(fact_2680_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_2681_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_2682_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [I2: A] :
                ( member(A,I2,A3)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).

% prod_mono_strict
tff(fact_2683_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3))
          <=> ? [X2: A] :
                ( member(A,X2,A3)
                & dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2)),aa(A,B,F2,X2)) ) ) ) ) ).

% even_prod_iff
tff(fact_2684_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_2685_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_2686_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_2687_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$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_ew(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),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),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_2688_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xb: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xb,Xaa,Xba,Xc) = Y )
     => ( Y = $ite(aa(nat,$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_2689_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A2),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2690_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_2691_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,B4)) )
             => ( ! [A4: A] :
                    ( member(A,A4,A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,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),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B3)) ) ) ) ) ) ).

% prod_mono2
tff(fact_2692_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3),aa(A,B,F2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ) ) ).

% prod_diff1
tff(fact_2693_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_ex(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod
tff(fact_2694_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_2695_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_ey(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ).

% pochhammer_prod_rev
tff(fact_2696_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_2697_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => sums(A,power_power(A,C2),divide_divide(A,one_one(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_2698_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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_ez(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.in_pairs
tff(fact_2699_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fa(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_2700_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_ey(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_2701_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_2702_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,K),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( member(int,K,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
                & ~ 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),
                  ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),K)
                  & ~ 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,K,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_2703_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( member(int,Xb,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & member(int,Xaa,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xb)
                      & ~ 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),
                        ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2)),Xb)
                        & ~ 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),product_Pair(int,int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_2704_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_2705_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_2706_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D4)))),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D4)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F2),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(real,F2)) ) ) ).

% sum_pos_lt_pair
tff(fact_2707_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$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_fb(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_fb(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),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),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fd(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ) ).

% polyfun_diff
tff(fact_2708_atMost__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,set(A),set_ord_atMost(A),Xb) = aa(A,set(A),set_ord_atMost(A),Y) )
        <=> ( Xb = Y ) ) ) ).

% atMost_eq_iff
tff(fact_2709_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,K: A] :
          ( member(A,Ia,aa(A,set(A),set_ord_atMost(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),K) ) ) ).

% atMost_iff
tff(fact_2710_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_2711_finite__atMost,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% finite_atMost
tff(fact_2712_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_dt(nat,A)) ) ).

% summable_zero
tff(fact_2713_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Ia: nat,F2: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dy(nat,fun(fun(nat,A),fun(nat,A)),Ia),F2)) ) ).

% summable_single
tff(fact_2714_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_2715_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),Xb)),aa(A,set(A),set_ord_atMost(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% atMost_subset_iff
tff(fact_2716_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ej(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_2717_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3) = one_one(nat) )
      <=> ! [X2: A] :
            ( member(A,X2,A3)
           => ( aa(A,nat,F2,X2) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_2718_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fe(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_2719_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),collect(nat,P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_en(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_2720_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A3)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eo(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_2721_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_2722_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_2723_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_2724_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,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),G),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atMost_Suc
tff(fact_2725_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,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),G),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atMost_Suc
tff(fact_2726_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3))
      <=> ! [X2: A] :
            ( member(A,X2,A3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X2)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_2727_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_2728_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Y))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_2729_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_2730_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,power_power(A,C2))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_2731_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_2732_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_2733_int__prod,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: 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),F2),A3)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7121269368397514597t_prod(A,int),aTP_Lamp_cq(fun(A,nat),fun(A,int),F2)),A3) ).

% int_prod
tff(fact_2734_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_2735_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_2736_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_2737_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_ff(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_2738_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N3: nat,F2: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
           => summable(A,F2) ) ) ) ).

% summable_comparison_test'
tff(fact_2739_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test
tff(fact_2740_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_2741_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_2742_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B3: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N))
         => ( ! [N: nat] : aa(A,$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),N))),B3)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_2743_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = collect(A,aTP_Lamp_fg(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_2744_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),F2),Xb))
         => ( aa(real,$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_fi(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_2745_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( summable(A,F2)
           => ( summable(A,G)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),suminf(A,G)) ) ) ) ) ).

% suminf_le
tff(fact_2746_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N3)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N3)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_2747_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_2748_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_atMost(nat),K) ).

% lessThan_Suc_atMost
tff(fact_2749_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ej(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_2750_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_2751_atMost__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,K)),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% atMost_Suc
tff(fact_2752_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dz(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_2753_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_2754_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_fj(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_fl(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% prod.triangle_reindex_eq
tff(fact_2755_finite__nat__iff__bounded__le,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ? [K2: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_atMost(nat),K2)) ) ).

% finite_nat_iff_bounded_le
tff(fact_2756_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N2: nat] : aa(nat,A,F2,N2) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_2757_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_nonneg
tff(fact_2758_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_pos
tff(fact_2759_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_fm(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_2760_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_dx(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_2761_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_fn(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_fl(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.triangle_reindex
tff(fact_2762_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Mb: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_fo(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),Mb),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_2763_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_fp(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_2764_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_2765_prod__int__eq,axiom,
    ! [Ia: 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,Ia,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_bj(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Ia),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_2766_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Ia: nat] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,Ia))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ) ).

% suminf_pos2
tff(fact_2767_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2))
            <=> ? [I4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I4)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_2768_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xb: A] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),N))),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),Xb) ) ) ) ).

% suminf_le_const
tff(fact_2769_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fq(fun(nat,A),fun(A,fun(nat,A)),F2),Xb))
         => ( aa(real,$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_fq(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_2770_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xb: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),N))),Xb)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_2771_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_2772_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => summable(A,power_power(A,C2)) ) ) ).

% summable_geometric
tff(fact_2773_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),F2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_2774_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cf(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% sum.atMost_Suc_shift
tff(fact_2775_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Ia: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_atMost(nat),Ia)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,Ia))) ) ).

% sum_telescope
tff(fact_2776_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,D2: fun(nat,A)] :
          ( ! [X2: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fr(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fr(fun(nat,A),fun(A,fun(nat,A)),D2),X2)),aa(nat,set(nat),set_ord_atMost(nat),Nb))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
             => ( aa(nat,A,C2,I4) = aa(nat,A,D2,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_2777_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% prod.atMost_Suc_shift
tff(fact_2778_prod__int__plus__eq,axiom,
    ! [Ia: 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,Ia,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_bj(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Ia),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J)))) ).

% prod_int_plus_eq
tff(fact_2779_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_fs(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_fu(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_2780_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_fv(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_fx(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_2781_binomial__maximum_H,axiom,
    ! [Nb: nat,K: 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)),K)),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_2782_binomial__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_mono
tff(fact_2783_binomial__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( 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),K)) ) ) ) ).

% binomial_antimono
tff(fact_2784_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_2785_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( aa(set(nat),$o,finite_finite2(nat),I5)
           => ( ! [N: nat] :
                  ( member(nat,N,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),I5)),suminf(A,F2)) ) ) ) ) ).

% sum_le_suminf
tff(fact_2786_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,F2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,fun(nat,A)),F2),K))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_2787_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_2788_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: nat] :
          ( ! [W: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),C2),W)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_2789_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( ! [X2: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fr(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
             => ( aa(nat,A,C2,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_2790_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,I2)) )
       => ( aa(real,real,ln_ln(real),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),F2),I5)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_fz(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_2791_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cf(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% sum.atMost_shift
tff(fact_2792_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),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),F2),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),F2),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_2793_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% prod.atMost_shift
tff(fact_2794_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)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_2795_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_fj(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_gb(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% sum.triangle_reindex_eq
tff(fact_2796_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat] :
          ( summable(A,F2)
         => ( ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,M2)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),suminf(A,F2)) ) ) ) ).

% sum_less_suminf
tff(fact_2797_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),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),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_2798_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_strict_mono
tff(fact_2799_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_2800_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fq(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_fq(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F2,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_2801_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fq(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gc(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fq(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_2802_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E2: real] :
          ( summable(A,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ~ ! [N8: nat] :
                  ~ ! [M: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),M)
                     => ! [N4: nat] : aa(real,$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),F2),set_or1337092689740270186AtMost(nat,M,N4)))),E2) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_2803_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( summable(A,F2)
           => ? [N8: nat] :
              ! [N4: nat] :
                ( aa(nat,$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_et(fun(nat,A),fun(nat,fun(nat,A)),F2),N4)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_2804_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,I2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Z)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_gd(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_2805_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,$o),ord_less(real),R2),R0)
           => ( ! [N: nat] : aa(real,$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,N))),aa(nat,real,power_power(real,R0),N))),M4)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_ge(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_2806_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_2807_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_gf(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
              & ( aa(nat,A,C2,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_2808_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_gf(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_2809_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: 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_fb(fun(nat,A),fun(A,fun(nat,A)),C2),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_fb(fun(nat,A),fun(A,fun(nat,A)),C2),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_fb(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_2810_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: 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_fb(fun(nat,A),fun(A,fun(nat,A)),C2),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_fb(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_fb(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% polyfun_linear_factor
tff(fact_2811_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_2812_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N3: nat,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,aa(nat,nat,suc,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N)))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_2813_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat,Ia: nat] :
          ( summable(A,F2)
         => ( ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M2)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Ia)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,Ia))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),suminf(A,F2)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_2814_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_fn(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_gb(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.triangle_reindex
tff(fact_2815_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,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_cn(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% sum.in_pairs_0
tff(fact_2816_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Mb: nat,A2: fun(nat,A),Nb: nat,B2: fun(nat,A),Xb: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I2)
             => ( aa(nat,A,A2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$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_fb(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_fb(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_gh(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_2817_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,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_ez(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% prod.in_pairs_0
tff(fact_2818_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: A] :
          ( ! [X2: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fr(fun(nat,A),fun(A,fun(nat,A)),C2),X2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X2: nat] :
                ( member(nat,X2,set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C2,X2) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_2819_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
     => ( ! [K3: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,K3),L3))
           => ( ( ~ ( member(int,K3,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L3,aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,K3,aa(num,int,numeral_numeral(int),bit0(one2)))),divide_divide(int,L3,aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K3),L3) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_2820_polynomial__product__nat,axiom,
    ! [Mb: nat,A2: fun(nat,nat),Nb: nat,B2: fun(nat,nat),Xb: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I2)
         => ( aa(nat,nat,A2,I2) = 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_gi(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_gi(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_gk(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_2821_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_2822_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),
            ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Mb)
            & ~ 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_2823_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gl(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2824_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gn(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_go(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_2825_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_gp(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_2826_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_2827_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$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_fb(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_fb(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),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),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_gr(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_2828_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_2829_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C2: fun(nat,A),Nb: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ? [M7: real] :
            ! [Z3: A] :
              ( aa(real,$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_fh(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),aa(nat,set(nat),set_ord_atMost(nat),Nb)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_2830_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_gs(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_2831_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_gt(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_2832_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_gu(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_2833_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_2834_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_gv(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2835_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_2836_binomial__1,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(nat,nat,suc,zero_zero(nat))) = Nb ).

% binomial_1
tff(fact_2837_binomial__0__Suc,axiom,
    ! [K: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2838_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_2839_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_2840_binomial__n__0,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2841_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
     => ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2842_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_2843_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R2)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R2)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_2844_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_2845_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K)) ) ).

% zero_less_binomial
tff(fact_2846_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_2847_choose__mult,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),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),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K))) ) ) ) ).

% choose_mult
tff(fact_2848_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_gw(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_2849_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_gx(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_2850_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_gx(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_2851_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$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),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2852_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% binomial_le_pow2
tff(fact_2853_choose__reduce__nat,axiom,
    ! [Nb: nat,K: 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)),K)
       => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2854_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(Nb),K)) = 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),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2855_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_gy(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_2856_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_gz(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_2857_binomial__addition__formula,axiom,
    ! [Nb: nat,K: 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,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2858_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_ha(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ).

% binomial
tff(fact_2859_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_hb(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% binomial_ring
tff(fact_2860_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_hc(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_2861_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_hd(real,fun(nat,real),Xb)) ) ).

% summable_arctan_series
tff(fact_2862_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_2863_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_he(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_2864_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_hg(fun(nat,A),fun(A,fun(nat,A)),C2),Xb),suminf(A,aa(A,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))) ) ) ).

% diffs_equiv
tff(fact_2865_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_2866_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_2867_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_2868_of__nat__id,axiom,
    ! [Nb: nat] : aa(nat,nat,semiring_1_of_nat(nat),Nb) = Nb ).

% of_nat_id
tff(fact_2869_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_2870_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_2871_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_2872_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2873_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ).

% abs_numeral
tff(fact_2874_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_2875_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_2876_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_2877_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_2878_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_2879_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_2880_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: A,K: A] :
          ( dvd_dvd(A,aa(A,A,abs_abs(A),Mb),K)
        <=> dvd_dvd(A,Mb,K) ) ) ).

% abs_dvd_iff
tff(fact_2881_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: A,K: A] :
          ( dvd_dvd(A,Mb,aa(A,A,abs_abs(A),K))
        <=> dvd_dvd(A,Mb,K) ) ) ).

% dvd_abs_iff
tff(fact_2882_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_2883_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_2884_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: $o] : aa(A,A,abs_abs(A),aa($o,A,zero_neq_one_of_bool(A),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% abs_bool_eq
tff(fact_2885_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_2886_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_2887_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_2888_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_2889_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% abs_neg_numeral
tff(fact_2890_abs__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).

% abs_neg_one
tff(fact_2891_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K: nat] : aa(nat,A,gbinomial(A,zero_zero(A)),aa(nat,nat,suc,K)) = zero_zero(A) ) ).

% gbinomial_0(2)
tff(fact_2892_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_2893_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_2894_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_hh(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs
tff(fact_2895_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_2896_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_2897_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_2898_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_2899_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,$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_hh(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs_ge_zero
tff(fact_2900_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_2901_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_2902_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_2903_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_2904_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_2905_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2906_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_2907_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),Xb) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( Xb = Y )
            | ( Xb = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_2908_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => dvd_dvd(A,L,K) ) ) ).

% dvd_if_abs_eq
tff(fact_2909_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_2910_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_2911_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_2912_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_2913_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ) ).

% abs_mult_less
tff(fact_2914_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_2915_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_2916_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_2917_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_2918_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_2919_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_2920_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_2921_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_2922_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_2923_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xb: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),E) )
         => ( Xb = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2924_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_2925_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$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),Y)),Xb) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb)) ) ) ) ).

% abs_mult_pos
tff(fact_2926_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_2927_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_2928_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_2929_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( divide_divide(A,aa(A,A,abs_abs(A),Xb),Y) = aa(A,A,abs_abs(A),divide_divide(A,Xb,Y)) ) ) ) ).

% abs_div_pos
tff(fact_2930_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_2931_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_2932_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_2933_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_2934_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_2935_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(A,$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),C2),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2936_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_2937_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_2938_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_2939_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_2940_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) = 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),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_2941_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F2,N))),aa(nat,real,G,N)) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_hi(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_2942_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_2943_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_2944_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_2945_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_2946_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)
         => aa(A,$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),K))),K)),aa(nat,A,gbinomial(A,A2),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_2947_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A2),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_2948_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A2),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_2949_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_2950_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_le_square_iff
tff(fact_2951_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% Suc_times_gbinomial
tff(fact_2952_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Mb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),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)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2953_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_hj(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_2954_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2955_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$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),P,X3),aa(nat,A,power_power(A,X3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P,aa(A,A,abs_abs(A),Xb)),aa(nat,A,power_power(A,Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_2956_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_2957_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_2958_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
         => ( aa(A,$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_2959_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_2960_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% gbinomial_factors
tff(fact_2961_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xb: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xb,I2)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Xb),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,I2)),B2))),Delta) )
             => aa(B,$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_hk(fun(A,B),fun(fun(A,B),fun(A,B)),Xb),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2962_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A2)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_2963_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2964_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,K5: real,C2: fun(nat,A)] :
          ( aa(real,$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_hl(fun(nat,A),fun(A,fun(nat,A)),C2),X3)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_hm(A,fun(fun(nat,A),fun(nat,A)),Xb),C2)) ) ) ) ).

% termdiff_converges
tff(fact_2965_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,A2: A,Xb: A,Y: 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_hn(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),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_ho(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) ) ).

% gbinomial_partial_sum_poly
tff(fact_2966_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hp(nat,fun(nat,A),K)),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),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2967_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_2968_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_2969_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_2970_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: 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),Y)),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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
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_hq(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,Y: 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_hn(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),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_hr(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_2973_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_he(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_2974_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)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [Y2: 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),Y2))),D4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_2975_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_hs(real,fun(nat,real),Xb)) ) ).

% monoseq_arctan_series
tff(fact_2976_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_hd(real,fun(nat,real),Xb)) ) ) ).

% arctan_series
tff(fact_2977_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)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [Y2: 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),Y2))),D4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2978_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_2979_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_2980_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_2981_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( dvd_dvd(int,Xb,one_one(int))
    <=> ( aa(int,int,abs_abs(int),Xb) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2982_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_2983_zdvd__antisym__abs,axiom,
    ! [A2: int,B2: int] :
      ( dvd_dvd(int,A2,B2)
     => ( dvd_dvd(int,B2,A2)
       => ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) ) ) ) ).

% zdvd_antisym_abs
tff(fact_2984_arctan__monotone,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xb)),arctan(Y)) ) ).

% arctan_monotone
tff(fact_2985_arctan__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xb)),arctan(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% arctan_less_iff
tff(fact_2986_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_2987_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S2)
    <=> ! [M3: int] :
        ? [N2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M3),aa(int,int,abs_abs(int),N2))
          & member(int,N2,S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_2988_infinite__int__iff__unbounded,axiom,
    ! [S2: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S2)
    <=> ! [M3: int] :
        ? [N2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M3),aa(int,int,abs_abs(int),N2))
          & member(int,N2,S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_2989_zabs__def,axiom,
    ! [Ia: int] :
      aa(int,int,abs_abs(int),Ia) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ia),zero_zero(int)),aa(int,int,uminus_uminus(int),Ia),Ia) ).

% zabs_def
tff(fact_2990_dvd__imp__le__int,axiom,
    ! [Ia: int,D2: int] :
      ( ( Ia != zero_zero(int) )
     => ( dvd_dvd(int,D2,Ia)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),Ia)) ) ) ).

% dvd_imp_le_int
tff(fact_2991_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_2992_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero(int) )
     => ( 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_2993_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,Mb)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
                & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2994_decr__lemma,axiom,
    ! [D2: int,Xb: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,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))),D2))),Z) ) ).

% decr_lemma
tff(fact_2995_incr__lemma,axiom,
    ! [D2: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),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))),D2))) ) ).

% incr_lemma
tff(fact_2996_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arctan_ubound
tff(fact_2997_nat__ivt__aux,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2998_arctan__lbound,axiom,
    ! [Y: 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(Y)) ).

% arctan_lbound
tff(fact_2999_arctan__bounded,axiom,
    ! [Y: 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(Y))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% arctan_bounded
tff(fact_3000_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_3001_arctan__add,axiom,
    ! [Xb: real,Y: 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),Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),arctan(Xb)),arctan(Y)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_3002_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M2: nat,N: nat] :
              ( aa(nat,$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)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI1
tff(fact_3003_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M2: nat,N: nat] :
              ( aa(nat,$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)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI2
tff(fact_3004_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [M3: nat,N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M3)),aa(nat,A,X5,N2)) )
            | ! [M3: nat,N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,M3)) ) ) ) ) ).

% monoseq_def
tff(fact_3005_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,aa(nat,nat,suc,N2)))
            | ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N2))),aa(nat,A,X5,N2)) ) ) ) ).

% monoseq_Suc
tff(fact_3006_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N))),aa(nat,A,X5,N))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI2
tff(fact_3007_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,aa(nat,nat,suc,N)))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI1
tff(fact_3008_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] :
          aa(nat,A,gbinomial(A,A2),K) = $ite(K = zero_zero(nat),one_one(A),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_ht(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_3009_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa(nat,nat,binomial(Nb),K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K),
        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))),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),divide_divide(nat,set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),one_one(nat)),Nb,one_one(nat)),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_3010_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),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,K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),Mb)),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,Nb,Mb))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))))) ) ).

% modulo_int_unfold
tff(fact_3011_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),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,K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Mb,Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,Mb,Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,Nb,Mb)))))) ) ).

% divide_int_unfold
tff(fact_3012_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_3013_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_3014_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_3015_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_3016_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_3017_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_3018_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_3019_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_3020_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_3021_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_3022_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3023_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_3024_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,Nb) )
    <=> ( pred_numeral(K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_3025_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa(nat,nat,suc,Nb) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( Nb = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_3026_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_3027_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_3028_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_3029_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_3030_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_3031_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_3032_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_3033_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),Nb) ) ).

% less_numeral_Suc
tff(fact_3034_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(bit1(K)) = aa(num,nat,numeral_numeral(nat),bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_3035_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_3036_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Nb) ) ).

% le_numeral_Suc
tff(fact_3037_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K)),Nb) ).

% diff_numeral_Suc
tff(fact_3038_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_3039_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( dvd_dvd(int,L,aa(int,int,aa(int,fun(int,int),times_times(int),K),sgn_sgn(int,R2)))
    <=> ( dvd_dvd(int,L,K)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3040_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( dvd_dvd(int,L,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R2)),K))
    <=> ( dvd_dvd(int,L,K)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3041_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),sgn_sgn(int,R2)),K)
    <=> ( dvd_dvd(int,L,K)
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3042_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,R2)),L),K)
    <=> ( dvd_dvd(int,L,K)
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3043_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),Nb)) ).

% max_numeral_Suc
tff(fact_3044_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_3045_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_3046_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_3047_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_3048_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_3049_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_3050_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_3051_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_3052_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_3053_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_3054_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_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_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_3062_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),sgn_sgn(A,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_3063_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_3064_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_3065_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L3: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L3)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_sgnE
tff(fact_3066_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_3067_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_3068_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_3069_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)
         => dvd_dvd(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,Mb)) ) ) ).

% fact_dvd
tff(fact_3070_numeral__eq__Suc,axiom,
    ! [K: num] : aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_3071_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_3072_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_3073_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_3074_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)
       => dvd_dvd(nat,Mb,semiring_char_0_fact(nat,Nb)) ) ) ).

% dvd_fact
tff(fact_3075_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_3076_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))) ) ).

% fact_fact_dvd_fact
tff(fact_3077_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_3078_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_3079_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_3080_lessThan__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_lessThan(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),pred_numeral(K)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_3081_atMost__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_atMost(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_3082_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_3083_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_3084_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_3085_zsgn__def,axiom,
    ! [Ia: int] :
      sgn_sgn(int,Ia) = $ite(
        Ia = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ia),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_3086_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_3087_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),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,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),aa(nat,nat,binomial(Nb),K)) = semiring_char_0_fact(nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_3088_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_3089_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))),semiring_char_0_fact(A,Nb)) ) ) ).

% choose_dvd
tff(fact_3090_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_bk(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_3091_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = 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,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3092_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_bk(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_3093_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_3094_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_3095_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = 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,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))) ) ) ) ).

% binomial_fact
tff(fact_3096_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) = 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),K))) ) ) ) ).

% fact_binomial
tff(fact_3097_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = divide_divide(A,comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_3098_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_hu(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_3099_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ? [B9: real] : aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_hv(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),B9),divide_divide(real,aa(nat,real,power_power(real,H),Nb),semiring_char_0_fact(real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_3100_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_3101_sin__coeff__def,axiom,
    ! [X: nat] :
      sin_coeff(X) = $ite(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_3102_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)
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T6))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T6)),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_hx(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,T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3103_sin__paired,axiom,
    ! [Xb: real] : sums(real,aTP_Lamp_hy(real,fun(nat,real),Xb),sin(real,Xb)) ).

% sin_paired
tff(fact_3104_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ia(A,fun(A,fun(nat,A)),Xb),Y),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_3105_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_3106_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_3107_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_3108_exp__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Y)) ) ).

% exp_less_mono
tff(fact_3109_exp__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% exp_less_cancel_iff
tff(fact_3110_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_3111_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_3112_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_3113_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_3114_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3115_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_3116_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_3117_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_3118_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_3119_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_3120_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_3121_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = 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),Y)) ) ).

% scaleR_right_distrib
tff(fact_3122_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_3123_exp__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xb)),exp(real,Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% exp_less_cancel
tff(fact_3124_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_3125_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X3: real] : exp(real,X3) = Y ) ).

% exp_total
tff(fact_3126_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_3127_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_3128_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xb)),exp(A,Y)) = exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ).

% mult_exp_exp
tff(fact_3129_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xb)),exp(A,Y)) ) ) ) ).

% exp_add_commuting
tff(fact_3130_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_3131_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),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,Y),Xaa)) ) ).

% scaleR_left.add
tff(fact_3132_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_3133_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C2)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3134_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_3135_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_3136_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_3137_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_3138_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_3139_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(real,$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),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_3140_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_3141_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V2: real,A2: A,Xb: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V2)),A2) = Xb )
        <=> $ite(V2 = zero_zero(real),Xb = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V2),Xb)) ) ) ).

% vector_fraction_eq_iff
tff(fact_3142_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,U: real,V2: real,A2: A] :
          ( ( Xb = aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V2)),A2) )
        <=> $ite(V2 = zero_zero(real),Xb = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V2),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_3143_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),E2)),C2)),D2) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3144_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),E2)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3145_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( exp(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aTP_Lamp_ib(fun(A,B),fun(A,B),F2)),I5) ) ) ) ).

% exp_sum
tff(fact_3146_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_3147_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_3148_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_3149_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_3150_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_3151_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_3152_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_3153_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_3154_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_3155_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D2)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3156_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(real,$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),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3157_ln__ge__iff,axiom,
    ! [Xb: real,Y: 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),Y),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_3158_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_3159_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_3160_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_3161_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_3162_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_3163_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_3164_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_3165_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_3166_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_3167_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_3168_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_3169_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_3170_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_3171_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: 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))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),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,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3172_sin__monotone__2pi,axiom,
    ! [Y: 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))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3173_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_3174_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_3175_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_3176_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R2) )
        & $ite(
            aa(int,$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_3177_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: 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))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_3178_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ? [T6: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T6)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),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_ic(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),T6),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_3179_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)
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T6)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),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_ic(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),T6),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_3180_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),A1) ) )
       => ( ! [Q4: int] :
              ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q4),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q4),A22) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q4),R3) )
               => ( ( 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),Q4),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_3181_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),K2) ) )
        | ? [L4: int,K2: int,Q5: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q5),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K2 = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L4) ) )
        | ? [R5: int,L4: int,K2: int,Q5: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa(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))
            & ( K2 = 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_3182_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ie(A,fun(A,fun(nat,A)),Xb),Y),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_3183_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ig(A,fun(A,fun(nat,A)),Xb),Y),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_3184_sincos__total__2pi,axiom,
    ! [Xb: real,Y: 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,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T6: real] :
            ( 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(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xb = cos(real,T6) )
               => ( Y != sin(real,T6) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3185_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_3186_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_ih(A,fun(nat,A),Xb))) ) ).

% exp_first_two_terms
tff(fact_3187_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_3188_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_3189_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3190_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_3191_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_3192_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_3193_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_3194_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_3195_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_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_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_3206_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_3207_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_3208_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_3209_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_3210_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_3211_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),W2)) ) ).

% inverse_eq_divide_numeral
tff(fact_3212_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = divide_divide(A,one_one(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_3213_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_3214_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_3215_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_3216_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_3217_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_3218_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_3219_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_3220_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_3221_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_3222_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_3223_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_3224_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_3225_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_3226_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_3227_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_3228_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_3229_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_3230_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_3231_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_3232_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_3233_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_3234_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_3235_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_3236_inverse__numeral__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),one2)) = aa(num,A,numeral_numeral(A),one2) ) ) ).

% inverse_numeral_1
tff(fact_3237_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_3238_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_3239_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_3240_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_3241_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_3242_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_3243_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_3244_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_3245_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_3246_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_3247_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_3248_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3249_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_3250_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_3251_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_3252_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_3253_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_3254_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_3255_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_3256_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_3257_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_3258_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_3259_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_3260_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = 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,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),sin(A,Y))) ) ).

% sin_add
tff(fact_3261_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_3262_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_3263_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y))) = divide_divide(A,cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3264_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = 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),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3265_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = 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),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3266_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_3267_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_3268_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_3269_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_3270_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_3271_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = 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,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% cos_add
tff(fact_3272_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = 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,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% cos_diff
tff(fact_3273_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_3274_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] : aa(A,$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,N)))),Xb) ) ) ).

% reals_Archimedean
tff(fact_3275_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_3276_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3277_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: 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)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xb)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3278_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Xb: A,C2: A,Y: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Mb),Xb)),C2) = Y )
          <=> ( 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)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_3279_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Y: A,Xb: A,C2: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Mb),Xb)),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),C2)) = Xb ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_3280_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3281_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3282_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3283_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3284_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_3285_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_3286_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_3287_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_3288_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_3289_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D4: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D4),E)
         => ( aa(real,$o,P,D4)
           => aa(real,$o,P,E) ) )
     => ( ! [N: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_3290_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
    <=> ? [N2: nat] :
          ( ( N2 != 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),N2)))
          & 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),N2))),E2) ) ) ).

% real_arch_inverse
tff(fact_3291_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D4: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D4),E)
         => ( aa(real,$o,P,D4)
           => aa(real,$o,P,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$o,P,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),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_3292_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_3293_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N))),Xb) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3294_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_3295_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_3296_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3297_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3298_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3299_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3300_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_3301_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3302_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3303_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3304_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_3305_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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,Y)),cos(real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3306_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),pi)
      & ( sin(real,Y3) = sin(real,Xb) )
      & ( cos(real,Y3) = cos(real,Xb) ) ) ).

% sincos_principal_value
tff(fact_3307_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_3308_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_3309_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_3310_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [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),Y),aa(real,real,tan(real),X3)) ) ) ).

% lemma_tan_total
tff(fact_3311_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_3312_tan__total,axiom,
    ! [Y: 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) = Y )
      & ! [Y2: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,tan(real),Y2) = Y ) )
         => ( Y2 = X3 ) ) ) ).

% tan_total
tff(fact_3313_tan__monotone,axiom,
    ! [Y: 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))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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),Y)),aa(real,real,tan(real),Xb)) ) ) ) ).

% tan_monotone
tff(fact_3314_tan__monotone_H,axiom,
    ! [Y: 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))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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),Y),Xb)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3315_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: 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))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3316_lemma__tan__total1,axiom,
    ! [Y: 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) = Y ) ) ).

% lemma_tan_total1
tff(fact_3317_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_3318_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_3319_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_3320_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_3321_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_3322_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_3323_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_3324_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_3325_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_3326_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_3327_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_3328_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [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) = Y ) ) ) ).

% tan_total_pos
tff(fact_3329_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_3330_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: 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))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3331_tan__mono__le,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3332_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_3333_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_3334_arctan__unique,axiom,
    ! [Xb: real,Y: 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) = Y )
         => ( arctan(Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_3335_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_3336_arctan,axiom,
    ! [Y: 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(Y))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),arctan(Y)) = Y ) ) ).

% arctan
tff(fact_3337_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),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),Y)),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_ii(A,fun(A,fun(nat,fun(nat,A))),Xb),Y),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_3338_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_ij(A,fun(nat,A),Xb))) ) ).

% exp_first_term
tff(fact_3339_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_3340_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,K: 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_ik(A,fun(nat,A),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_il(A,fun(nat,fun(nat,A)),Xb),K))) ) ).

% exp_first_terms
tff(fact_3341_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))
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T6)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),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_im(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),T6),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_3342_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)
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T6)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),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_im(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),T6),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_3343_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_in(A,fun(nat,A),Xb),sinh(A,Xb)) ) ).

% sinh_converges
tff(fact_3344_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_io(A,fun(nat,A),Xb),cosh(A,Xb)) ) ).

% cosh_converges
tff(fact_3345_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T6: real] :
            ( 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(real),T6),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(cos(real,T6),sin(real,T6)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3346_sinh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% sinh_real_less_iff
tff(fact_3347_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_3348_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_3349_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_3350_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_3351_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3352_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_3353_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_3354_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = 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,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_3355_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = 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,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_3356_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_3357_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_3358_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_3359_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: 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)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_3360_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: 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),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_3361_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_3362_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_3363_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cosh(A,Xb) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = 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),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),Xb)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_3364_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)),insert2(A),one_one(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_3365_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_3366_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_3367_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_3368_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_3369_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_3370_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_3371_divide__int__def,axiom,
    ! [K: int,L: int] :
      divide_divide(int,K,L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(int,L,K)))))) ) ).

% divide_int_def
tff(fact_3372_arctan__def,axiom,
    ! [Y: real] : arctan(Y) = the(real,aTP_Lamp_ip(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_3373_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_3374_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_3375_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_3376_nat__int,axiom,
    ! [Nb: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),Nb)) = Nb ).

% nat_int
tff(fact_3377_nat__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_3378_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_3379_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_3380_nat__of__bool,axiom,
    ! [P: $o] : aa(int,nat,nat2,aa($o,int,zero_neq_one_of_bool(int),(P))) = aa($o,nat,zero_neq_one_of_bool(nat),(P)) ).

% nat_of_bool
tff(fact_3381_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_3382_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_3383_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_3384_nat__0__iff,axiom,
    ! [Ia: int] :
      ( ( aa(int,nat,nat2,Ia) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_3385_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_3386_nat__neg__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_3387_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_3388_log__less__cancel__iff,axiom,
    ! [A2: real,Xb: real,Y: 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)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_3389_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_3390_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_3391_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_3392_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_3393_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_3394_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_3395_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_3396_log__le__cancel__iff,axiom,
    ! [A2: real,Xb: real,Y: 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)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_3397_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_3398_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_3399_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_3400_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_3401_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_3402_diff__nat__numeral,axiom,
    ! [V2: num,V4: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),aa(num,nat,numeral_numeral(nat),V4)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),V4))) ).

% diff_nat_numeral
tff(fact_3403_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xb)),Nb) = aa(int,nat,nat2,Y) )
    <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_3404_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa(int,nat,nat2,Y) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xb)),Nb) )
    <=> ( Y = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_3405_nat__abs__dvd__iff,axiom,
    ! [K: int,Nb: nat] :
      ( dvd_dvd(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),Nb)
    <=> dvd_dvd(int,K,aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_abs_dvd_iff
tff(fact_3406_dvd__nat__abs__iff,axiom,
    ! [Nb: nat,K: int] :
      ( dvd_dvd(nat,Nb,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),Nb),K) ) ).

% dvd_nat_abs_iff
tff(fact_3407_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_3408_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_3409_nat__numeral__diff__1,axiom,
    ! [V2: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V2)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_3410_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_3411_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_3412_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_3413_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_3414_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_3415_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_3416_nat__numeral__as__int,axiom,
    ! [X: num] : aa(num,nat,numeral_numeral(nat),X) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X)) ).

% nat_numeral_as_int
tff(fact_3417_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ).

% nat_mono
tff(fact_3418_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: 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)),Z5)
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_3419_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_13: nat] : aa(nat,$o,P,X_13)
    <=> ! [X2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X2)
         => aa(nat,$o,P,aa(int,nat,nat2,X2)) ) ) ).

% all_nat
tff(fact_3420_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_13: nat] : aa(nat,$o,P,X_13)
    <=> ? [X2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X2)
          & aa(nat,$o,P,aa(int,nat,nat2,X2)) ) ) ).

% ex_nat
tff(fact_3421_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_3422_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_3423_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_3424_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_3425_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_3426_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_3427_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_3428_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_3429_int__minus,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) = aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(nat,int,semiring_1_of_nat(int),Mb)))) ).

% int_minus
tff(fact_3430_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_3431_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_3432_nat__times__as__int,axiom,
    ! [X: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_3433_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_3434_nat__minus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_3435_nat__div__as__int,axiom,
    ! [X: nat,Xa: nat] : divide_divide(nat,X,Xa) = aa(int,nat,nat2,divide_divide(int,aa(nat,int,semiring_1_of_nat(int),X),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_3436_nat__mod__as__int,axiom,
    ! [X: nat,Xa: nat] : modulo_modulo(nat,X,Xa) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_mod_as_int
tff(fact_3437_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_3438_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_3439_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_3440_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_3441_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_3442_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_3443_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_3444_split__nat,axiom,
    ! [P: fun(nat,$o),Ia: int] :
      ( aa(nat,$o,P,aa(int,nat,nat2,Ia))
    <=> ( ! [N2: nat] :
            ( ( Ia = aa(nat,int,semiring_1_of_nat(int),N2) )
           => aa(nat,$o,P,N2) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ia),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_3445_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K) ) ) ).

% le_nat_iff
tff(fact_3446_nat__add__distrib,axiom,
    ! [Z: int,Z5: 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)),Z5)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_3447_nat__mult__distrib,axiom,
    ! [Z: int,Z5: 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),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_3448_Suc__as__int,axiom,
    ! [X: nat] : aa(nat,nat,suc,X) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X)),one_one(int))) ).

% Suc_as_int
tff(fact_3449_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: 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)),Y)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_3450_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z5),Z)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_3451_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_3452_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_3453_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_3454_log__mult,axiom,
    ! [A2: real,Xb: real,Y: 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)),Y)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_3455_log__divide,axiom,
    ! [A2: real,Xb: real,Y: 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)),Y)
           => ( aa(real,real,log(A2),divide_divide(real,Xb,Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_3456_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_3457_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_3458_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_3459_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_3460_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_3461_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_3462_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: 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),Z5)) = 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),Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_3463_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_3464_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z5)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z5),zero_zero(int)),
        aa(int,nat,nat2,Z),
        $let(
          d: int,
          d:= aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z5),
          $ite(aa(int,$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_3465_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_3466_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_3467_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_3468_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K))) ) ).

% of_int_of_nat
tff(fact_3469_nat__dvd__iff,axiom,
    ! [Z: int,Mb: nat] :
      ( 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),dvd_dvd(int,Z,aa(nat,int,semiring_1_of_nat(int),Mb)),Mb = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_3470_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( 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_3471_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_3472_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_3473_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_3474_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_3475_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_3476_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: 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)),K)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),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_3477_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),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),K))) = 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_3478_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_3479_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: 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)),K)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Nb) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),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_3480_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),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),K))) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_3481_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_3482_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_3483_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_3484_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_3485_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_3486_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V2)),Xb) ) ) ).

% numeral_le_floor
tff(fact_3487_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_3488_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_3489_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_3490_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% floor_less_numeral
tff(fact_3491_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_3492_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_3493_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% ceiling_le_numeral
tff(fact_3494_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_3495_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_3496_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_3497_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V2)),Xb) ) ) ).

% numeral_less_ceiling
tff(fact_3498_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_3499_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_3500_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_add_numeral
tff(fact_3501_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_3502_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_3503_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_3504_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_3505_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$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),V2)),one_one(A))),Xb) ) ) ).

% numeral_less_floor
tff(fact_3506_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$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),V2)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_3507_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_3508_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_3509_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$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),V2)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_3510_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,Xb))
        <=> aa(A,$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),V2)),one_one(A))),Xb) ) ) ).

% numeral_le_ceiling
tff(fact_3511_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),Xb) ) ) ).

% neg_numeral_le_floor
tff(fact_3512_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_neg_numeral
tff(fact_3513_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_3514_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),Xb) ) ) ).

% neg_numeral_less_ceiling
tff(fact_3515_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$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),V2))),one_one(A))),Xb) ) ) ).

% neg_numeral_less_floor
tff(fact_3516_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$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),V2))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_3517_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$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),V2))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_3518_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,Xb))
        <=> aa(A,$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),V2))),one_one(A))),Xb) ) ) ).

% neg_numeral_le_ceiling
tff(fact_3519_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_3520_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_3521_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% floor_less_cancel
tff(fact_3522_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,Xb)) ) ) ).

% ceiling_mono
tff(fact_3523_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_3524_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% ceiling_less_cancel
tff(fact_3525_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_3526_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_3527_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$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,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))) ) ).

% le_floor_add
tff(fact_3528_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_3529_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_3530_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_3531_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_3532_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_3533_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_3534_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$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),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_3535_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_3536_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_3537_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_3538_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_3539_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_3540_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_3541_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_3542_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_3543_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_3544_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_3545_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_3546_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_3547_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_3548_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
             => aa(int,$o,P,I4) ) ) ) ).

% floor_split
tff(fact_3549_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_3550_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_3551_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_3552_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_3553_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_3554_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_3555_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_3556_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),aa(int,A,ring_1_of_int(A),I4)) )
             => aa(int,$o,P,I4) ) ) ) ).

% ceiling_split
tff(fact_3557_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_3558_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_3559_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_3560_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_3561_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_3562_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( aa(A,$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,P3,Q2)))),Q2)),P3) ) ) ).

% floor_divide_lower
tff(fact_3563_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( aa(A,$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),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P3,Q2)))),Q2)) ) ) ).

% ceiling_divide_upper
tff(fact_3564_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P3,Q2)))),one_one(A))),Q2)) ) ) ).

% floor_divide_upper
tff(fact_3565_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_3566_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P3: A] :
          ( aa(A,$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,P3,Q2)))),one_one(A))),Q2)),P3) ) ) ).

% ceiling_divide_lower
tff(fact_3567_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_3568_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_3569_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_3570_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: 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),K)),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),K))),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),K),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_3571_powr__int,axiom,
    ! [Xb: real,Ia: 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),Ia)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ia),aa(nat,real,power_power(real,Xb),aa(int,nat,nat2,Ia)),divide_divide(real,one_one(real),aa(nat,real,power_power(real,Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Ia))))) ) ) ).

% powr_int
tff(fact_3572_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),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,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3573_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: 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)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K))),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),K),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_3574_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_3575_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_3576_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_3577_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_3578_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_3579_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_3580_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_3581_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_3582_log__powr__cancel,axiom,
    ! [A2: real,Y: 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,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_3583_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_3584_powr__less__mono2__neg,axiom,
    ! [A2: real,Xb: real,Y: 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),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Y,A2)),powr(real,Xb,A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_3585_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_3586_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_3587_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_3588_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_3589_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_3590_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_3591_powr__less__mono2,axiom,
    ! [A2: real,Xb: real,Y: 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),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_3592_powr__mono2_H,axiom,
    ! [A2: real,Xb: real,Y: 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),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,A2)),powr(real,Xb,A2)) ) ) ) ).

% powr_mono2'
tff(fact_3593_powr__inj,axiom,
    ! [A2: real,Xb: real,Y: 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,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_3594_gr__one__powr,axiom,
    ! [Xb: real,Y: 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)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),powr(real,Xb,Y)) ) ) ).

% gr_one_powr
tff(fact_3595_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_3596_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_3597_powr__less__iff,axiom,
    ! [B2: real,Xb: real,Y: 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,Y)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_3598_less__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: 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,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_3599_log__less__iff,axiom,
    ! [B2: real,Xb: real,Y: 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)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_3600_less__log__iff,axiom,
    ! [B2: real,Xb: real,Y: 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),Y),aa(real,real,log(B2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_3601_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_3602_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3603_arcsin__less__mono,axiom,
    ! [Xb: real,Y: 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),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_3604_powr__le__iff,axiom,
    ! [B2: real,Xb: real,Y: 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,Y)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_3605_le__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: 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,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_3606_log__le__iff,axiom,
    ! [B2: real,Xb: real,Y: 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)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_3607_le__log__iff,axiom,
    ! [B2: real,Xb: real,Y: 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),Y),aa(real,real,log(B2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_3608_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_3609_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = $ite(aa(A,$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,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_3610_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_3611_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_3612_log__add__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: 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)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_3613_add__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: 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),Y),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,Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_3614_minus__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: 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),Y),aa(real,real,log(B2),Xb)) = aa(real,real,log(B2),divide_divide(real,powr(real,B2,Y),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3615_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_3616_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_3617_log__minus__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: 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)),Y) = 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),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_3618_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_3619_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = $ite(aa(A,$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,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_3620_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_3621_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_3622_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_3623_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($o,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb))) ).

% Suc_0_xor_eq
tff(fact_3624_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($o,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb))) ).

% xor_Suc_0_eq
tff(fact_3625_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_iq(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_3626_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_3627_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_3628_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_3629_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_3630_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_3631_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_3632_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_add
tff(fact_3633_the__elem__eq,axiom,
    ! [A: $tType,Xb: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) = Xb ).

% the_elem_eq
tff(fact_3634_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_3635_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),bit1(Y)) ).

% xor_nat_numerals(1)
tff(fact_3636_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Y))) = aa(num,nat,numeral_numeral(nat),bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_3637_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_3638_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_3639_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_3640_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_3641_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: 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(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(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),Y)))) ) ).

% xor_numerals(4)
tff(fact_3642_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: 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(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(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),Y)))) ) ).

% xor_numerals(6)
tff(fact_3643_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_3644_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_3645_floor__real__def,axiom,
    ! [Xb: real] : archim6421214686448440834_floor(real,Xb) = the(int,aTP_Lamp_ir(real,fun(int,$o),Xb)) ).

% floor_real_def
tff(fact_3646_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_3647_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),
            ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Mb) != ~ 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_3648_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),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2))) ) ).

% xor_one_eq
tff(fact_3649_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),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2))) ) ).

% one_xor_eq
tff(fact_3650_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_3651_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_3652_floor__rat__def,axiom,
    ! [Xb: rat] : archim6421214686448440834_floor(rat,Xb) = the(int,aTP_Lamp_is(rat,fun(int,$o),Xb)) ).

% floor_rat_def
tff(fact_3653_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F2: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_it(set(A),fun(fun(A,B),fun(A,$o)),I5),F2))),F4)
           => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ia),bot_bot(set(A))))) = $ite(member(A,Ia,I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,Ia)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3654_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_3655_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_3656_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_3657_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_3658_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P3,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_3659_sum_Oeq__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1027152243600224163dd_sum(A,B,P3,I5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),P3),I5) ) ) ) ).

% sum.eq_sum
tff(fact_3660_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_3661_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Y))) = aa(num,nat,numeral_numeral(nat),bit1(Y)) ).

% or_nat_numerals(2)
tff(fact_3662_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( groups1027152243600224163dd_sum(A,B,P3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ia),I5)) = $ite(member(A,Ia,I5),groups1027152243600224163dd_sum(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P3,Ia)),groups1027152243600224163dd_sum(A,B,P3,I5))) ) ) ) ).

% sum.insert'
tff(fact_3663_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_3664_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),bit1(Y)) ).

% or_nat_numerals(1)
tff(fact_3665_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: 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(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(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),Y)))) ) ).

% or_numerals(7)
tff(fact_3666_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: 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(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(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),Y)))) ) ).

% or_numerals(6)
tff(fact_3667_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: 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(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(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),Y)))) ) ).

% or_numerals(4)
tff(fact_3668_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xb),Y)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_3669_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)
           => ! [T6: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T6)
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S3),T6) ) ) ) ) ).

% obtain_pos_sum
tff(fact_3670_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_3671_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_3672_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_3673_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_3674_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,collect(B,aa(set(B),fun(B,$o),aTP_Lamp_iu(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_3675_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_iv(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_3676_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G,X3) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S2) = groups1027152243600224163dd_sum(A,B,G,T3) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3677_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G,X3) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T3) = groups1027152243600224163dd_sum(A,B,G,S2) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3678_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,H,I2) = zero_zero(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S2)
                 => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S2) = groups1027152243600224163dd_sum(A,B,H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3679_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G,X3) = zero_zero(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S2)
                 => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T3) = groups1027152243600224163dd_sum(A,B,H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3680_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xb: A,Y: 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),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( Xb = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_3681_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_iv(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_3682_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_iu(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_iu(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_3683_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),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2))) ) ).

% one_or_eq
tff(fact_3684_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),dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2))) ) ).

% or_one_eq
tff(fact_3685_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: 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),Y),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),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_3686_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_it(set(A),fun(fun(A,B),fun(A,$o)),I5),F2)))
         => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ia),bot_bot(set(A))))) = $ite(member(A,Ia,I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,Ia)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ).

% sum_diff1'
tff(fact_3687_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),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb))) ).

% Suc_0_or_eq
tff(fact_3688_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),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb))) ).

% or_Suc_0_eq
tff(fact_3689_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),
            ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Mb)
            | ~ 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_3690_normalize__negative,axiom,
    ! [Q2: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q2),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_3691_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_bk(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_3692_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_3693_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M3)
         => ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N2)
             => 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,M3)),aa(nat,real,X5,N2)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_3694_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_3695_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or7035219750837199246ssThan(nat,L,U)) ).

% finite_atLeastLessThan
tff(fact_3696_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,L: A,U: A] :
          ( member(A,Ia,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Ia)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_3697_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_3698_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ia: A,J: A,Mb: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Ia,J)),set_or7035219750837199246ssThan(A,Mb,Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Ia)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),Ia)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Nb) ) ) ) ) ).

% ivl_subset
tff(fact_3699_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_3700_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_3701_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_3702_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ia: A,Nb: A,Mb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),Nb)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,Ia,Mb)),set_or7035219750837199246ssThan(A,Ia,Nb)) = set_or7035219750837199246ssThan(A,Nb,Mb) ) ) ) ).

% ivl_diff
tff(fact_3703_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_3704_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)),insert2(nat),Mb),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_3705_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.op_ivl_Suc
tff(fact_3706_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.op_ivl_Suc
tff(fact_3707_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A2 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_3708_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_3709_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_3710_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_3711_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_3712_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Nb)
          & aa(nat,$o,P,M3) )
    <=> ? [X2: nat] :
          ( member(nat,X2,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X2) ) ) ).

% ex_nat_less_eq
tff(fact_3713_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Nb)
         => aa(nat,$o,P,M3) )
    <=> ! [X2: nat] :
          ( member(nat,X2,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X2) ) ) ).

% all_nat_less_eq
tff(fact_3714_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_3715_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_3716_atLeastLessThan0,axiom,
    ! [Mb: nat] : set_or7035219750837199246ssThan(nat,Mb,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_3717_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cf(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_3718_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_cg(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_3719_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_3720_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ei(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_3721_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D2)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_3722_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D2)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_3723_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$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),P3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_3724_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,P3: nat,F2: fun(nat,A)] :
          ( aa(nat,$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),P3)
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,Mb,P3))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,Mb,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,Nb,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_3725_size__list__estimation,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F2,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(list(A),nat,size_list(A,F2),Xs)) ) ) ).

% size_list_estimation
tff(fact_3726_size__list__estimation_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F2,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(list(A),nat,size_list(A,F2),Xs)) ) ) ).

% size_list_estimation'
tff(fact_3727_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X3)),aa(A,nat,G,X3)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_list(A,F2),Xs)),aa(list(A),nat,size_list(A,G),Xs)) ) ).

% size_list_pointwise
tff(fact_3728_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$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),P3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_3729_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)),insert2(nat),Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_3730_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_3731_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3732_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3733_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_3734_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,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),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_3735_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_3736_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_3737_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_3738_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,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),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_3739_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_3740_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_3741_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% sum.last_plus
tff(fact_3742_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% prod.last_plus
tff(fact_3743_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$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_ck(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_Suc_diff'
tff(fact_3744_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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_iw(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% sum.atLeastLessThan_rev
tff(fact_3745_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)),insert2(nat),Nb),set_or7035219750837199246ssThan(nat,Mb,Nb)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_3746_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_ix(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_fu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% sum.nested_swap
tff(fact_3747_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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_iy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% prod.atLeastLessThan_rev
tff(fact_3748_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_iz(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_fx(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% prod.nested_swap
tff(fact_3749_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: 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_ja(fun(nat,A),fun(nat,fun(nat,A)),G),K)),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),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% sum.nat_group
tff(fact_3750_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: 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_jb(fun(nat,A),fun(nat,fun(nat,A)),G),K)),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),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% prod.nat_group
tff(fact_3751_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_3752_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_3753_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.head_if
tff(fact_3754_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.head_if
tff(fact_3755_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P3: int,Q2: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% normalize_denom_pos
tff(fact_3756_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_3757_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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_cj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3758_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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_er(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_3759_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_ex(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_prod
tff(fact_3760_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      set_or7035219750837199246ssThan(nat,Mb,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),pred_numeral(K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),pred_numeral(K)),set_or7035219750837199246ssThan(nat,Mb,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_3761_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_3762_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N6: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M3)
                 => ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,M3,N2)))),E3) ) ) ) ) ).

% summable_Cauchy
tff(fact_3763_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M7: nat] :
              ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M)
               => ! [N4: nat] :
                    ( aa(nat,$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,M)),aa(nat,A,X5,N4)))),E2) ) ) ) ) ) ).

% CauchyD
tff(fact_3764_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M9: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N)
                     => aa(real,$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)))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_3765_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M8: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N2)
                     => aa(real,$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,M3)),aa(nat,A,X5,N2)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_3766_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A,K: nat] :
          ( sums(A,F2,S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_jc(fun(nat,A),fun(nat,fun(nat,A)),F2),K),S) ) ) ) ).

% sums_group
tff(fact_3767_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: 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),Y),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),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ) ) ).

% OR_upper
tff(fact_3768_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)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3769_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),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),K),Nb)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))) ) ) ) ).

% fact_split
tff(fact_3770_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jd(nat,fun(nat,fun(nat,A)),K),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_3771_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_je(A,fun(nat,fun(nat,A)),A2),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_3772_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A2),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jf(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_3773_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),semiring_char_0_fact(A,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jf(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_3774_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_3775_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_3776_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_jg(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3777_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I2)),aa(nat,A,A2,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I2)) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),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_jh(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_3778_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A2,I2)),aa(nat,nat,A2,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I2)) ) )
       => 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_ji(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_3779_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_3780_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_3781_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_3782_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_3783_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_3784_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( collect(A,P) = bot_bot(set(A)) )
    <=> ( P = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_3785_is__singleton__the__elem,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the_elem(A,A3)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_3786_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_3787_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3))
          <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(set(A),nat,finite_card(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_jj(set(A),fun(fun(A,B),fun(A,$o)),A3),F2)))) ) ) ) ).

% even_sum_iff
tff(fact_3788_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_3789_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_3790_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_3791_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_3792_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,finite_card(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_3793_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_3794_card_Oinfinite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_3795_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_3796_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_3797_is__singletonI,axiom,
    ! [A: $tType,Xb: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_3798_card__0__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
      <=> ( A3 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_3799_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ~ member(A,Xb,A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_3800_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( member(A,A2,A3)
     => ( ~ member(A,A2,B3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_3801_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_3802_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_3803_is__singleton__altdef,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_3804_n__subsets,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(set(A)),nat,finite_card(set(A)),collect(set(A),aa(nat,fun(set(A),$o),aTP_Lamp_jk(set(A),fun(nat,fun(set(A),$o)),A3),K))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A3)),K) ) ) ).

% n_subsets
tff(fact_3805_card__subset__eq,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
       => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,finite_card(A),B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_3806_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ? [B9: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B9)
          & ( aa(set(A),nat,finite_card(A),B9) = Nb )
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B9),A3) ) ) ).

% infinite_arbitrarily_large
tff(fact_3807_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B3: set(A),A3: set(B),R2: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( ! [A4: B] :
            ( member(B,A4,A3)
           => ? [B10: A] :
                ( member(A,B10,B3)
                & aa(A,$o,aa(B,fun(A,$o),R2,A4),B10) ) )
       => ( ! [A12: B,A23: B,B4: A] :
              ( member(B,A12,A3)
             => ( member(B,A23,A3)
               => ( member(A,B4,B3)
                 => ( aa(A,$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),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_3808_card__insert__le,axiom,
    ! [A: $tType,A3: set(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3))) ).

% card_insert_le
tff(fact_3809_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_3810_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S: set(A),Ta: set(B),R: fun(A,fun(B,$o)),K: 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),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S),R),X3))) = aa(B,nat,K,X3) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_jl(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),K),Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_3811_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_at(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(nat,nat,power_power(nat,aa(set(A),nat,finite_card(A),A3)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_3812_is__singletonI_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [X3: A,Y3: A] :
            ( member(A,X3,A3)
           => ( member(A,Y3,A3)
             => ( X3 = Y3 ) ) )
       => is_singleton(A,A3) ) ) ).

% is_singletonI'
tff(fact_3813_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)) )
    <=> ? [X2: A] :
          ( member(A,X2,S2)
          & ? [Xa3: A] :
              ( member(A,Xa3,S2)
              & ( X2 != Xa3 )
              & ! [Xb4: A] :
                  ( member(A,Xb4,S2)
                 => ( ( Xb4 = X2 )
                    | ( Xb4 = Xa3 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_3814_card__eq__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = zero_zero(nat) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% card_eq_0_iff
tff(fact_3815_card__ge__0__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
     => aa(set(A),$o,finite_finite2(A),A3) ) ).

% card_ge_0_finite
tff(fact_3816_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(member(A,Xb,A3),aa(set(A),nat,finite_card(A),A3),aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_insert_if
tff(fact_3817_card__Suc__eq__finite,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B11: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B7),B11) )
          & ~ member(A,B7,B11)
          & ( aa(set(A),nat,finite_card(A),B11) = K )
          & aa(set(A),$o,finite_finite2(A),B11) ) ) ).

% card_Suc_eq_finite
tff(fact_3818_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C5: nat] :
      ( ! [G2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G2),F4)
         => ( aa(set(A),$o,finite_finite2(A),G2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G2)),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_3819_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Nb: nat,S2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),S2))
     => ~ ! [T5: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T5),S2)
           => ( ( aa(set(A),nat,finite_card(A),T5) = Nb )
             => ~ aa(set(A),$o,finite_finite2(A),T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_3820_card__seteq,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B3)),aa(set(A),nat,finite_card(A),A3))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_3821_card__mono,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ).

% card_mono
tff(fact_3822_card__less__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))
         => aa(nat,$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)),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_3823_card__le__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))
         => aa(nat,$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)),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_3824_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_3825_card__1__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) )
     => ~ ! [X3: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_3826_psubset__card__mono,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ).

% psubset_card_mono
tff(fact_3827_card__less__Suc2,axiom,
    ! [M4: set(nat),Ia: nat] :
      ( ~ member(nat,zero_zero(nat),M4)
     => ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_jm(set(nat),fun(nat,fun(nat,$o)),M4),Ia))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_jn(set(nat),fun(nat,fun(nat,$o)),M4),Ia))) ) ) ).

% card_less_Suc2
tff(fact_3828_card__less__Suc,axiom,
    ! [M4: set(nat),Ia: nat] :
      ( member(nat,zero_zero(nat),M4)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_jm(set(nat),fun(nat,fun(nat,$o)),M4),Ia)))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_jn(set(nat),fun(nat,fun(nat,$o)),M4),Ia))) ) ) ).

% card_less_Suc
tff(fact_3829_card__less,axiom,
    ! [M4: set(nat),Ia: nat] :
      ( member(nat,zero_zero(nat),M4)
     => ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_jn(set(nat),fun(nat,fun(nat,$o)),M4),Ia))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3830_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_jo(fun(A,nat),fun(A,nat),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,finite_card(A),A3)) ).

% sum_Suc
tff(fact_3831_subset__card__intvl__is__intvl,axiom,
    ! [A3: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A3),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))))
     => ( A3 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_3832_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T3: set(B),R: fun(A,fun(B,$o)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),T3)
       => ( ! [X3: B] :
              ( member(B,X3,T3)
             => ( aa(set(A),nat,finite_card(A),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S2),R),X3))) = K ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_jl(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T3),R)),S2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T3)) ) ) ) ) ).

% sum_multicount
tff(fact_3833_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),K5: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K5),aa(A,B,F2,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)) ) ) ).

% sum_bounded_below
tff(fact_3834_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),K5) )
         => aa(B,$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),F2),A3)),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),A3))),K5)) ) ) ).

% sum_bounded_above
tff(fact_3835_card__gt__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
    <=> ( ( A3 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% card_gt_0_iff
tff(fact_3836_card__1__singleton__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X2: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_3837_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
     => ? [B4: A,B9: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B4),B9) )
          & ~ member(A,B4,B9)
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & ( ( K = zero_zero(nat) )
           => ( B9 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_3838_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B11: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B7),B11) )
          & ~ member(A,B7,B11)
          & ( aa(set(A),nat,finite_card(A),B11) = K )
          & ( ( K = zero_zero(nat) )
           => ( B11 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_3839_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X2: A] :
            ( member(A,X2,A3)
           => ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( X2 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_3840_card__le__Suc__iff,axiom,
    ! [A: $tType,Nb: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),A3))
    <=> ? [A7: A,B11: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),B11) )
          & ~ member(A,A7,B11)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),B11))
          & aa(set(A),$o,finite_finite2(A),B11) ) ) ).

% card_le_Suc_iff
tff(fact_3841_card__Diff1__le,axiom,
    ! [A: $tType,A3: set(A),Xb: A] : aa(nat,$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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ).

% card_Diff1_le
tff(fact_3842_card__Diff__subset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ).

% card_Diff_subset
tff(fact_3843_card__psubset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3) ) ) ) ).

% card_psubset
tff(fact_3844_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => 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),A3)),aa(set(A),nat,finite_card(A),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ).

% diff_card_le_card_Diff
tff(fact_3845_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_au(set(A),fun(nat,fun(list(A),$o)),A3),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),A3))),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ) ).

% card_lists_length_le
tff(fact_3846_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,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aTP_Lamp_aw(nat,fun(A,$o),Nb)))),Nb) ) ) ).

% card_roots_unity
tff(fact_3847_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_3848_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_bk(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_bk(nat,nat)),S2)) ).

% card_sum_le_nat_sum
tff(fact_3849_card__nth__roots,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != 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),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_jp(complex,fun(nat,fun(complex,$o)),C2),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_3850_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),collect(complex,aTP_Lamp_cs(nat,fun(complex,$o),Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_3851_is__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
     => ~ ! [X3: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_3852_is__singleton__def,axiom,
    ! [A: $tType,A3: set(A)] :
      ( is_singleton(A,A3)
    <=> ? [X2: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_3853_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)) )
    <=> ? [X2: A,Y4: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y4),bot_bot(set(A)))) )
          & ( X2 != Y4 ) ) ) ).

% card_2_iff
tff(fact_3854_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)) )
    <=> ? [X2: A,Y4: A,Z6: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z6),bot_bot(set(A))))) )
          & ( X2 != Y4 )
          & ( Y4 != Z6 )
          & ( X2 != Z6 ) ) ) ).

% card_3_iff
tff(fact_3855_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),aa(set(A),nat,finite_card(A),A3))
     => ( A3 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_3856_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( member(A,Xb,A3)
       => ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_3857_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_3858_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( member(A,Xb,A3)
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_3859_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))
    <=> ( aa(set(A),$o,finite_finite2(A),A3)
        & member(A,Xb,A3) ) ) ).

% card_Diff1_less_iff
tff(fact_3860_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),Xb: A,Y: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( member(A,Xb,A3)
       => ( member(A,Y,A3)
         => 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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_Diff2_less
tff(fact_3861_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( member(A,Xb,A3)
       => 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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Diff1_less
tff(fact_3862_card__Diff__singleton__if,axiom,
    ! [A: $tType,A3: 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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) = $ite(member(A,Xb,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)),aa(set(A),nat,finite_card(A),A3)) ).

% card_Diff_singleton_if
tff(fact_3863_card__Diff__singleton,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] :
      ( member(A,Xb,A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_3864_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),Nb: B,K: nat] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),Nb) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),K)
           => ( aa(B,$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),F2),A3)),aa(nat,B,power_power(B,Nb),K)) ) ) ) ) ).

% prod_le_power
tff(fact_3865_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),K5) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),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),A3))),K5)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3866_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A3: set(A),F2: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),divide_divide(B,K5,aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3)))) )
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),K5) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_3867_card__insert__le__m1,axiom,
    ! [A: $tType,Nb: nat,Y: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))
       => aa(nat,$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)),insert2(A),Xb),Y))),Nb) ) ) ).

% card_insert_le_m1
tff(fact_3868_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_gf(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ).

% polyfun_roots_card
tff(fact_3869_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C2: B] :
          ( aa(set(A),$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_jq(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C2)),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,C2),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,C2),aa(set(A),nat,finite_card(A),S2))) ) ) ) ).

% prod_gen_delta
tff(fact_3870_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_gf(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_gf(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ) ).

% polyfun_rootbound
tff(fact_3871_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A3))
       => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_jr(set(A),fun(nat,fun(list(A),$o)),A3),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bk(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_3872_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_3873_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(set(A),fun(list(A),$o),aTP_Lamp_js(nat,fun(set(A),fun(list(A),$o)),K),A3))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bk(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_3874_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_3875_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_3876_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_3877_distinct__swap,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$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,Ia,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),Ia)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_3878_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ).

% bit_0
tff(fact_3879_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) )
            & ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_3880_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_jr(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_3881_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) )
         => ( aa(nat,$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_3882_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_3883_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_3884_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_3885_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_3886_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_3887_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_3888_finite__distinct__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A3 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_3889_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) )
         => ( 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_3890_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_3891_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_3892_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_3893_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_3894_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),Ia: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$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),Ia) = aa(nat,A,nth(A,Xs),J) )
          <=> ( Ia = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_3895_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_3896_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_3897_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,$o),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X3) = Xb )
            & ! [Y2: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y2),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y2) = Xb ) )
               => ( Y2 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_3898_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3899_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_3900_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),M)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_3901_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_3902_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,Ia: 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)),insert2(A),aa(nat,A,nth(A,Xs),Ia)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,Ia,A2)) ) ) ).

% distinct_list_update
tff(fact_3903_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( aa(nat,$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_3904_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2)
         => ( aa(nat,$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_3905_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)
          <=> $ite(Nb = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2),aa(nat,$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_3906_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> $ite(Nb = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2),aa(nat,$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_3907_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)),insert2(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)),insert2(A),aa(nat,A,nth(A,Xs),Nb)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_3908_set__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)) = collect(list(A),aa(list(A),fun(list(A),$o),aTP_Lamp_jt(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ).

% set_n_lists
tff(fact_3909_distinct__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),union(A),Xs),Ys))
    <=> distinct(A,Ys) ) ).

% distinct_union
tff(fact_3910_card__Pow,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow2(A,A3)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Pow
tff(fact_3911_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V2: num,Nb: nat] : case_nat(A,A2,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Nb)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),Nb)) ).

% case_nat_add_eq_if
tff(fact_3912_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))),insert2(set(A)),Y5),bot_bot(set(set(A)))) )
    <=> ( ( X5 = bot_bot(set(A)) )
        & ( Y5 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_3913_Pow__empty,axiom,
    ! [A: $tType] : pow2(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_3914_Pow__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( member(set(A),A3,pow2(A,B3))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% Pow_iff
tff(fact_3915_PowI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => member(set(A),A3,pow2(A,B3)) ) ).

% PowI
tff(fact_3916_finite__Pow__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),pow2(A,A3))
    <=> aa(set(A),$o,finite_finite2(A),A3) ) ).

% finite_Pow_iff
tff(fact_3917_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V2: num] : case_nat(A,A2,F2,aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,F2,pred_numeral(V2)) ).

% case_nat_numeral
tff(fact_3918_Pow__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pow2(A,A3)),pow2(A,B3)) ) ).

% Pow_mono
tff(fact_3919_PowD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( member(set(A),A3,pow2(A,B3))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% PowD
tff(fact_3920_Pow__not__empty,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,A3) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_3921_Pow__top,axiom,
    ! [A: $tType,A3: set(A)] : member(set(A),A3,pow2(A,A3)) ).

% Pow_top
tff(fact_3922_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_ju(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_3923_Pow__bottom,axiom,
    ! [A: $tType,B3: set(A)] : member(set(A),bot_bot(set(A)),pow2(A,B3)) ).

% Pow_bottom
tff(fact_3924_Pow__def,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,A3) = collect(set(A),aTP_Lamp_ae(set(A),fun(set(A),$o),A3)) ).

% Pow_def
tff(fact_3925_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X22: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X22)) = aa(nat,A,F22,X22) ).

% old.nat.simps(5)
tff(fact_3926_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_3927_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_jv(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_3928_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_jw(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_3929_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_3930_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_jx(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc1
tff(fact_3931_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_jy(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc2
tff(fact_3932_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_3933_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_bk(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ).

% diff_Suc
tff(fact_3934_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xb: A,F2: fun(nat,A),Nb: nat] :
      case_nat(A,Xb,F2,Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_3935_binomial__def,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(Nb),K) = aa(set(set(nat)),nat,finite_card(set(nat)),collect(set(nat),aa(nat,fun(set(nat),$o),aTP_Lamp_jz(nat,fun(nat,fun(set(nat),$o)),Nb),K))) ).

% binomial_def
tff(fact_3936_length__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,Nb,Xs)) = aa(nat,nat,power_power(nat,aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_n_lists
tff(fact_3937_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),F1: A,F22: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P,case_nat(A,F1,F22,Nat))
    <=> ( ( ( Nat = zero_zero(nat) )
         => aa(A,$o,P,F1) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => aa(A,$o,P,aa(nat,A,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_3938_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),F1: A,F22: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P,case_nat(A,F1,F22,Nat))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ aa(A,$o,P,F1) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ aa(A,$o,P,aa(nat,A,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_3939_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V2: num,Nb: nat] :
      aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Nb)) = $let(
        pv: nat,
        pv:= pred_numeral(V2),
        aa(A,A,aa(nat,fun(A,A),F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb)),aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb))) ) ).

% rec_nat_add_eq_if
tff(fact_3940_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_3941_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_3942_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_3943_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_3944_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_3945_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_3946_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_3947_rec__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V2: num] :
      aa(nat,A,rec_nat(A,A2,F2),aa(num,nat,numeral_numeral(nat),V2)) = $let(
        pv: nat,
        pv:= pred_numeral(V2),
        aa(A,A,aa(nat,fun(A,A),F2,pv),aa(nat,A,rec_nat(A,A2,F2),pv)) ) ).

% rec_nat_numeral
tff(fact_3948_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_3949_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2))
        <=> ( ( Nb != zero_zero(nat) )
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2)),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_3950_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_3951_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_3952_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,Mb),K)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_3953_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_3954_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_bk(nat,nat),Nat) ).

% pred_def
tff(fact_3955_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_3956_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_ka(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% take_bit_sum
tff(fact_3957_old_Orec__nat__def,axiom,
    ! [A: $tType,X: A,Xa: fun(nat,fun(A,A)),Xb3: nat] : aa(nat,A,rec_nat(A,X,Xa),Xb3) = the(A,rec_set_nat(A,X,Xa,Xb3)) ).

% old.rec_nat_def
tff(fact_3958_rec__nat__0__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F2 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F2,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_3959_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero(nat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_3960_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xb),aa(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_3961_subset__CollectI,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),Q: fun(A,$o),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
     => ( ! [X3: A] :
            ( member(A,X3,B3)
           => ( aa(A,$o,Q,X3)
             => aa(A,$o,P,X3) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),B3),Q))),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% subset_CollectI
tff(fact_3962_subset__Collect__iff,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),A3),P)))
      <=> ! [X2: A] :
            ( member(A,X2,B3)
           => aa(A,$o,P,X2) ) ) ) ).

% subset_Collect_iff
tff(fact_3963_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      nat_prod_decode_aux(K,Mb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),K),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),Mb)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_3964_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_3965_card__UNION,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A3)
     => ( ! [X3: set(A)] :
            ( member(set(A),X3,A3)
           => aa(set(A),$o,finite_finite2(A),X3) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_kb(set(set(A)),int)),collect(set(set(A)),aTP_Lamp_kc(set(set(A)),fun(set(set(A)),$o),A3)))) ) ) ) ).

% card_UNION
tff(fact_3966_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_3967_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)),insert2(A),Xb),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_3968_Sup__lessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Y: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_lessThan(A),Y)) = Y ) ).

% Sup_lessThan
tff(fact_3969_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_3970_Sup__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atMost(A),Y)) = Y ) ).

% Sup_atMost
tff(fact_3971_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_3972_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_3973_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_3974_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_3975_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Xb,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_3976_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,Xb)) = Xb ) ) ) ).

% cSup_atLeastAtMost
tff(fact_3977_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)),insert2(A),Xb),bot_bot(set(A)))) = Xb ) ).

% cSup_singleton
tff(fact_3978_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Xb,Y)) = Xb ) ) ) ).

% Inf_atLeastAtMost
tff(fact_3979_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,Xb)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_3980_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)),insert2(A),Xb),bot_bot(set(A)))) = Xb ) ).

% cInf_singleton
tff(fact_3981_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Xb,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_3982_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,Xb)) = Xb ) ) ) ).

% cSup_atLeastLessThan
tff(fact_3983_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Xb,Y)) = Xb ) ) ) ).

% Inf_atLeastLessThan
tff(fact_3984_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,Xb)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_3985_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_3986_finite__Union,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A3)
     => ( ! [M7: set(A)] :
            ( member(set(A),M7,A3)
           => 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)),A3)) ) ) ).

% finite_Union
tff(fact_3987_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_3988_remove1__commute,axiom,
    ! [A: $tType,Xb: A,Y: A,Zs: list(A)] : remove1(A,Xb,remove1(A,Y,Zs)) = remove1(A,Y,remove1(A,Xb,Zs)) ).

% remove1_commute
tff(fact_3989_notin__set__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ~ member(A,Xb,aa(list(A),set(A),set2(A),remove1(A,Y,Xs))) ) ).

% notin_set_remove1
tff(fact_3990_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_3991_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,$o),ord_less_eq(A),A2),X3) )
         => ( ! [Y3: A] :
                ( ! [X: A] :
                    ( member(A,X,X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_3992_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_3993_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,$o),ord_less_eq(A),X3),A2) )
         => ( ! [Y3: A] :
                ( ! [X: A] :
                    ( member(A,X,X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_3994_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_3995_distinct__remove1,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => distinct(A,remove1(A,Xb,Xs)) ) ).

% distinct_remove1
tff(fact_3996_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,$o),ord_less_eq(A),A2),X3) )
           => ( ! [Y3: A] :
                  ( ! [X: A] :
                      ( member(A,X,X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X5) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_3997_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_3998_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_3999_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_4000_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_4001_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,$o),ord_less_eq(A),X3),A2) )
           => ( ! [Y3: A] :
                  ( ! [X: A] :
                      ( member(A,X,X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_4002_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_4003_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_4004_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_4005_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
         => ( ( X5 != bot_bot(set(A)) )
           => ~ ! [X3: A] :
                  ( member(A,X3,X5)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X3) ) ) ) ) ).

% less_cSupE
tff(fact_4006_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_4007_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_4008_finite__UnionD,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3))
     => aa(set(set(A)),$o,finite_finite2(set(A)),A3) ) ).

% finite_UnionD
tff(fact_4009_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_4010_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))
            <=> ! [X2: A] :
                  ( member(A,X2,X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X2) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_4011_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)
            <=> ! [X2: A] :
                  ( member(A,X2,X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),A2) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_4012_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_4013_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_4014_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_4015_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E2) )
           => aa(A,$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))),E2) ) ) ) ).

% cInf_asclose
tff(fact_4016_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),L))),E2) )
           => aa(A,$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))),E2) ) ) ) ).

% cSup_asclose
tff(fact_4017_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)),insert2(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_4018_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_4019_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)),insert2(A),Xb),bot_bot(set(A)))) = Xb ) ).

% ccpo_Sup_singleton
tff(fact_4020_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_4021_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_4022_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A3) = bot_bot(A) )
        <=> ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X2)
             => ? [Xa3: A] :
                  ( member(A,Xa3,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X2) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_4023_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A3) )
        <=> ! [X2: A] :
              ( member(A,X2,A3)
             => ( X2 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_4024_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A3) = bot_bot(A) )
        <=> ! [X2: A] :
              ( member(A,X2,A3)
             => ( X2 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_4025_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_4026_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_4027_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V2: A] :
          ( member(A,U,A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),U)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).

% Sup_upper2
tff(fact_4028_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2)
        <=> ! [X2: A] :
              ( member(A,X2,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B2) ) ) ) ).

% Sup_le_iff
tff(fact_4029_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A3: set(A)] :
          ( member(A,Xb,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).

% Sup_upper
tff(fact_4030_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(A,$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),A3)),Z) ) ) ).

% Sup_least
tff(fact_4031_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ! [A4: A] :
              ( member(A,A4,A3)
             => ? [X: A] :
                  ( member(A,X,B3)
                  & aa(A,$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),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ).

% Sup_mono
tff(fact_4032_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Xb: A] :
          ( ! [Y3: A] :
              ( member(A,Y3,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
         => ( ! [Y3: A] :
                ( ! [Z3: A] :
                    ( member(A,Z3,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A3) = Xb ) ) ) ) ).

% Sup_eqI
tff(fact_4033_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S2: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2))
        <=> ? [X2: A] :
              ( member(A,X2,S2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X2) ) ) ) ).

% less_Sup_iff
tff(fact_4034_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(A,$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),A3)) ) ) ).

% Inf_greatest
tff(fact_4035_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3))
        <=> ! [X2: A] :
              ( member(A,X2,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X2) ) ) ) ).

% le_Inf_iff
tff(fact_4036_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V2: A] :
          ( member(A,U,A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V2) ) ) ) ).

% Inf_lower2
tff(fact_4037_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A3: set(A)] :
          ( member(A,Xb,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),Xb) ) ) ).

% Inf_lower
tff(fact_4038_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( ! [B4: A] :
              ( member(A,B4,B3)
             => ? [X: A] :
                  ( member(A,X,A3)
                  & aa(A,$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),A3)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ).

% Inf_mono
tff(fact_4039_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Xb: A] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),I2) )
         => ( ! [Y3: A] :
                ( ! [I3: A] :
                    ( member(A,I3,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),I3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A3) = Xb ) ) ) ) ).

% Inf_eqI
tff(fact_4040_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S2: set(A),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2)
        <=> ? [X2: A] :
              ( member(A,X2,S2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),A2) ) ) ) ).

% Inf_less_iff
tff(fact_4041_empty__Union__conv,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) )
    <=> ! [X2: set(A)] :
          ( member(set(A),X2,A3)
         => ( X2 = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_4042_Union__empty__conv,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) = bot_bot(set(A)) )
    <=> ! [X2: set(A)] :
          ( member(set(A),X2,A3)
         => ( X2 = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_4043_Union__least,axiom,
    ! [A: $tType,A3: set(set(A)),C5: set(A)] :
      ( ! [X6: set(A)] :
          ( member(set(A),X6,A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),C5) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),C5) ) ).

% Union_least
tff(fact_4044_Union__upper,axiom,
    ! [A: $tType,B3: set(A),A3: set(set(A))] :
      ( member(set(A),B3,A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) ) ).

% Union_upper
tff(fact_4045_Union__subsetI,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] :
      ( ! [X3: set(A)] :
          ( member(set(A),X3,A3)
         => ? [Y2: set(A)] :
              ( member(set(A),Y2,B3)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Y2) ) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) ) ).

% Union_subsetI
tff(fact_4046_Inter__greatest,axiom,
    ! [A: $tType,A3: set(set(A)),C5: set(A)] :
      ( ! [X6: set(A)] :
          ( member(set(A),X6,A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),X6) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)) ) ).

% Inter_greatest
tff(fact_4047_Inter__lower,axiom,
    ! [A: $tType,B3: set(A),A3: set(set(A))] :
      ( member(set(A),B3,A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B3) ) ).

% Inter_lower
tff(fact_4048_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A3))
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xb)
             => ? [X2: A] :
                  ( member(A,X2,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X2) ) ) ) ) ).

% le_Sup_iff
tff(fact_4049_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),Xb)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y4)
             => ? [X2: A] :
                  ( member(A,X2,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4) ) ) ) ) ).

% Inf_le_iff
tff(fact_4050_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3) )
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).

% less_eq_Sup
tff(fact_4051_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ).

% Sup_subset_mono
tff(fact_4052_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),U) )
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),U) ) ) ) ).

% Inf_less_eq
tff(fact_4053_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ).

% Inf_superset_mono
tff(fact_4054_Union__mono,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),A3),B3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) ) ).

% Union_mono
tff(fact_4055_Inter__anti__mono,axiom,
    ! [A: $tType,B3: set(set(A)),A3: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),B3),A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3)) ) ).

% Inter_anti_mono
tff(fact_4056_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_4057_Inter__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(A)] :
      ( ! [X6: set(A)] :
          ( member(set(A),X6,A3)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),B3) )
     => ( ( A3 != bot_bot(set(set(A))) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B3) ) ) ).

% Inter_subset
tff(fact_4058_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).

% Inf_le_Sup
tff(fact_4059_finite__subset__Union,axiom,
    ! [A: $tType,A3: set(A),B12: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12))
       => ~ ! [F6: set(set(A))] :
              ( aa(set(set(A)),$o,finite_finite2(set(A)),F6)
             => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),F6),B12)
               => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F6)) ) ) ) ) ).

% finite_subset_Union
tff(fact_4060_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( strict_mono_on(A,B,F2,A3)
        <=> ! [R5: A,S6: A] :
              ( ( member(A,R5,A3)
                & member(A,S6,A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R5),S6) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R5)),aa(A,B,F2,S6)) ) ) ) ).

% strict_mono_on_def
tff(fact_4061_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [R3: A,S3: A] :
              ( member(A,R3,A3)
             => ( member(A,S3,A3)
               => ( aa(A,$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,F2,R3)),aa(A,B,F2,S3)) ) ) )
         => strict_mono_on(A,B,F2,A3) ) ) ).

% strict_mono_onI
tff(fact_4062_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A),R2: A,S: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( member(A,R2,A3)
           => ( member(A,S,A3)
             => ( aa(A,$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,F2,R2)),aa(A,B,F2,S)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_4063_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_12)
     => ( ! [X3: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N),X3)
           => ? [Y2: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N)),Y2)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N),X3),Y2) ) )
       => ? [F3: fun(nat,A)] :
          ! [N4: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N4),aa(nat,A,F3,N4))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N4),aa(nat,A,F3,N4)),aa(nat,A,F3,aa(nat,nat,suc,N4))) ) ) ) ).

% dependent_nat_choice
tff(fact_4064_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F2: fun(A,B),A3: set(A),Xb: A,Y: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( member(A,Xb,A3)
           => ( member(A,Y,A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_4065_card__partition,axiom,
    ! [A: $tType,C5: set(set(A)),K: nat] :
      ( aa(set(set(A)),$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))
       => ( ! [C4: set(A)] :
              ( member(set(A),C4,C5)
             => ( aa(set(A),nat,finite_card(A),C4) = K ) )
         => ( ! [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),K),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_4066_totally__bounded__Union,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [M4: set(set(A))] :
          ( aa(set(set(A)),$o,finite_finite2(set(A)),M4)
         => ( ! [S5: set(A)] :
                ( member(set(A),S5,M4)
               => topolo6688025880775521714ounded(A,S5) )
           => topolo6688025880775521714ounded(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),M4)) ) ) ) ).

% totally_bounded_Union
tff(fact_4067_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_4068_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_4069_IntI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,A3)
     => ( member(A,C2,B3)
       => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) ) ).

% IntI
tff(fact_4070_Int__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
    <=> ( member(A,C2,A3)
        & member(A,C2,B3) ) ) ).

% Int_iff
tff(fact_4071_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) ) ) ) ).

% le_inf_iff
tff(fact_4072_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
        <=> ( aa(A,$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),C2) ) ) ) ).

% inf.bounded_iff
tff(fact_4073_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_4074_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_4075_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_4076_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_4077_finite__Int,axiom,
    ! [A: $tType,F4: set(A),G3: set(A)] :
      ( ( aa(set(A),$o,finite_finite2(A),F4)
        | aa(set(A),$o,finite_finite2(A),G3) )
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),F4),G3)) ) ).

% finite_Int
tff(fact_4078_Int__subset__iff,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),A3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),B3) ) ) ).

% Int_subset_iff
tff(fact_4079_Int__insert__right__if1,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( member(A,A2,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ) ) ).

% Int_insert_right_if1
tff(fact_4080_Int__insert__right__if0,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ~ member(A,A2,A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ).

% Int_insert_right_if0
tff(fact_4081_insert__inter__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ).

% insert_inter_insert
tff(fact_4082_Int__insert__left__if1,axiom,
    ! [A: $tType,A2: A,C5: set(A),B3: set(A)] :
      ( member(A,A2,C5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) ) ) ).

% Int_insert_left_if1
tff(fact_4083_Int__insert__left__if0,axiom,
    ! [A: $tType,A2: A,C5: set(A),B3: set(A)] :
      ( ~ member(A,A2,C5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5) ) ) ).

% Int_insert_left_if0
tff(fact_4084_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_4085_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_4086_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_4087_Pow__Int__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A3)),pow2(A,B3)) ).

% Pow_Int_eq
tff(fact_4088_totally__bounded__empty,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => topolo6688025880775521714ounded(A,bot_bot(set(A))) ) ).

% totally_bounded_empty
tff(fact_4089_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_4090_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_4091_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),Xb))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_4092_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: 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)),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_4093_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: 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),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_4094_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A3: set(A),B2: A,B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),B3)) )
    <=> ( ~ member(A,B2,A3)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ) ).

% disjoint_insert(2)
tff(fact_4095_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B3: set(A),A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = bot_bot(set(A)) )
    <=> ( ~ member(A,A2,B3)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_4096_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)),B3) )
    <=> ( ~ member(A,A2,B3)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ) ) ) ).

% insert_disjoint(2)
tff(fact_4097_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)),B3) = bot_bot(set(A)) )
    <=> ( ~ member(A,A2,B3)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_4098_Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_4099_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_4100_root__0,axiom,
    ! [Xb: real] : aa(real,real,root(zero_zero(nat)),Xb) = zero_zero(real) ).

% root_0
tff(fact_4101_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: 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),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4102_Compl__disjoint,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_4103_Compl__disjoint2,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_4104_Diff__Compl,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Diff_Compl
tff(fact_4105_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_4106_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_4107_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_4108_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: 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),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_4109_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: 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),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_4110_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_4111_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_4112_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_4113_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_4114_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: 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),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_4115_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_4116_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: 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),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_4117_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_4118_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: 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),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_4119_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_4120_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: 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),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_4121_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),P: fun(A,$o),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_kd(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F2)),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_4122_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),F2: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_ke(fun(A,B),fun(fun(A,$o),fun(A,B)),F2),P)),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_4123_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_4124_sum__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_kf(fun(A,$o),fun(A,B),P)),A3) = 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)),A3),collect(A,P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_4125_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] : aa(A,$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)),A3),B3))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ).

% Sup_inter_less_eq
tff(fact_4126_Union__Int__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3))) ).

% Union_Int_subset
tff(fact_4127_Int__emptyI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X3: A] :
          ( member(A,X3,A3)
         => ~ member(A,X3,B3) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_4128_disjoint__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ! [X2: A] :
          ( member(A,X2,A3)
         => ~ member(A,X2,B3) ) ) ).

% disjoint_iff
tff(fact_4129_Int__empty__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B3) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_4130_Int__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_4131_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ! [X2: A] :
          ( member(A,X2,A3)
         => ! [Xa3: A] :
              ( member(A,Xa3,B3)
             => ( X2 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4132_Int__insert__left,axiom,
    ! [A: $tType,A2: A,B3: set(A),C5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C5) = $ite(member(A,A2,C5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) ).

% Int_insert_left
tff(fact_4133_Int__insert__right,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = $ite(member(A,A2,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) ).

% Int_insert_right
tff(fact_4134_Int__mono,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),B3: set(A),D5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),D5)
       => aa(set(A),$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)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),D5)) ) ) ).

% Int_mono
tff(fact_4135_Int__lower1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),A3) ).

% Int_lower1
tff(fact_4136_Int__lower2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),B3) ).

% Int_lower2
tff(fact_4137_Int__absorb1,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = B3 ) ) ).

% Int_absorb1
tff(fact_4138_Int__absorb2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = A3 ) ) ).

% Int_absorb2
tff(fact_4139_Int__greatest,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),B3)
       => aa(set(A),$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)),inf_inf(set(A)),A3),B3)) ) ) ).

% Int_greatest
tff(fact_4140_Int__Collect__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( ! [X3: A] :
            ( member(A,X3,A3)
           => ( aa(A,$o,P,X3)
             => aa(A,$o,Q,X3) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),collect(A,Q))) ) ) ).

% Int_Collect_mono
tff(fact_4141_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_4142_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Xb) ) ).

% inf_sup_ord(1)
tff(fact_4143_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Xb) ) ).

% inf_le1
tff(fact_4144_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Y) ) ).

% inf_le2
tff(fact_4145_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_4146_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_4147_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$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),C2),D2)) ) ) ) ).

% inf_mono
tff(fact_4148_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_4149_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_4150_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_4151_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_4152_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X3: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),Y3)),X3)
         => ( ! [X3: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),Y3)),Y3)
           => ( ! [X3: A,Y3: A,Z2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F2,Y3),Z2)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F2,Xb),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_4153_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Xb ) ) ) ).

% le_iff_inf
tff(fact_4154_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_4155_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_4156_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Xb ) ) ) ).

% inf_absorb1
tff(fact_4157_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_4158_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$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),C2) ) ) ) ).

% inf.boundedE
tff(fact_4159_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ) ) ).

% inf.boundedI
tff(fact_4160_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$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),Y),Z)) ) ) ) ).

% inf_greatest
tff(fact_4161_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_4162_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_4163_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_4164_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_4165_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_4166_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.coboundedI1
tff(fact_4167_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.coboundedI2
tff(fact_4168_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.strict_coboundedI2
tff(fact_4169_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.strict_coboundedI1
tff(fact_4170_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_4171_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% inf.strict_boundedE
tff(fact_4172_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_4173_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_4174_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_4175_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_4176_IntE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => ~ ( member(A,C2,A3)
         => ~ member(A,C2,B3) ) ) ).

% IntE
tff(fact_4177_IntD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => member(A,C2,A3) ) ).

% IntD1
tff(fact_4178_IntD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => member(A,C2,B3) ) ).

% IntD2
tff(fact_4179_Int__assoc,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) ).

% Int_assoc
tff(fact_4180_Int__absorb,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),A3) = A3 ).

% Int_absorb
tff(fact_4181_Int__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3) ).

% Int_commute
tff(fact_4182_Int__left__absorb,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Int_left_absorb
tff(fact_4183_Int__left__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5)) ).

% Int_left_commute
tff(fact_4184_Diff__Int__distrib2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) ).

% Diff_Int_distrib2
tff(fact_4185_Diff__Int__distrib,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),B3)) ).

% Diff_Int_distrib
tff(fact_4186_Diff__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) ).

% Diff_Diff_Int
tff(fact_4187_Diff__Int2,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5)),B3) ).

% Diff_Int2
tff(fact_4188_Int__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C5)) ).

% Int_Diff
tff(fact_4189_Int__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_kg(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% Int_def
tff(fact_4190_Int__Collect,axiom,
    ! [A: $tType,Xb: A,A3: set(A),P: fun(A,$o)] :
      ( member(A,Xb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P)))
    <=> ( member(A,Xb,A3)
        & aa(A,$o,P,Xb) ) ) ).

% Int_Collect
tff(fact_4191_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),collect(A,P)),collect(A,Q)) ).

% Collect_conj_eq
tff(fact_4192_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_4193_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_4194_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B3: set(A),A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B3)),A2) = bot_bot(A) )
        <=> ! [X2: A] :
              ( member(A,X2,B3)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),A2) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_4195_Diff__triv,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = A3 ) ) ).

% Diff_triv
tff(fact_4196_Int__Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_4197_Union__disjoint,axiom,
    ! [A: $tType,C5: set(set(A)),A3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)),A3) = bot_bot(set(A)) )
    <=> ! [X2: set(A)] :
          ( member(set(A),X2,C5)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X2),A3) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_4198_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_4199_Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Diff_eq
tff(fact_4200_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_4201_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_4202_totally__bounded__subset,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A),T3: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S2)
           => topolo6688025880775521714ounded(A,T3) ) ) ) ).

% totally_bounded_subset
tff(fact_4203_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: 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),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_4204_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: 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),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_4205_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: 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),K)) = aa(nat,real,power_power(real,aa(real,real,root(Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_4206_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_4207_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% inf_shunt
tff(fact_4208_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] :
                  ( member(A,X3,A3)
                 => ( member(A,Y3,A3)
                   => member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3),A3) ) )
             => member(A,aa(set(A),A,complete_Inf_Inf(A),A3),A3) ) ) ) ) ).

% finite_Inf_in
tff(fact_4209_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_4210_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_4211_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_4212_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_4213_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ) ).

% disjoint_eq_subset_Compl
tff(fact_4214_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))),insert2(set(A)),Xb),F4))
           => ! [Xa2: set(A)] :
                ( member(set(A),Xa2,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Xb),F4))
               => ( ( X3 != Xa2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa2) = 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_4215_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(A),fun(A,B),aTP_Lamp_kh(fun(A,B),fun(set(A),fun(A,B)),G),B3)),A3) ) ) ) ).

% sum.inter_restrict
tff(fact_4216_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_ki(fun(A,B),fun(set(A),fun(A,B)),G),B3)),A3) ) ) ) ).

% prod.inter_restrict
tff(fact_4217_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_4218_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_4219_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_4220_root__abs__power,axiom,
    ! [Nb: nat,Y: 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,Y),Nb))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_4221_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,H,I2) = zero_zero(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T3))
                   => ( aa(A,B,G,I2) = zero_zero(B) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3))
                     => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T3) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_4222_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: 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),K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),K),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_4223_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ) ) ).

% sum.Int_Diff
tff(fact_4224_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))) ) ) ) ).

% prod.Int_Diff
tff(fact_4225_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,H,I2) = one_one(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T3))
                   => ( aa(A,B,G,I2) = one_one(B) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3))
                     => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T3) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_4226_card__Diff__subset__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ).

% card_Diff_subset_Int
tff(fact_4227_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_kj(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A3) = 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)),A3),collect(A,P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P))))) ) ) ) ).

% sum.If_cases
tff(fact_4228_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_kk(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A3) = 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)),A3),collect(A,P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P))))) ) ) ) ).

% prod.If_cases
tff(fact_4229_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_4230_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_4231_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_4232_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_4233_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_4234_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: 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)),Y)
       => ( ( aa(nat,real,power_power(real,Y),Nb) = Xb )
         => ( aa(real,real,root(Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4235_dvd__partition,axiom,
    ! [A: $tType,C5: set(set(A)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5))
     => ( ! [X3: set(A)] :
            ( member(set(A),X3,C5)
           => dvd_dvd(nat,K,aa(set(A),nat,finite_card(A),X3)) )
       => ( ! [X3: set(A)] :
              ( member(set(A),X3,C5)
             => ! [Xa2: set(A)] :
                  ( member(set(A),Xa2,C5)
                 => ( ( X3 != Xa2 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa2) = bot_bot(set(A)) ) ) ) )
         => dvd_dvd(nat,K,aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5))) ) ) ) ).

% dvd_partition
tff(fact_4236_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A3: set(A),F2: fun(A,B),B2: B] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3),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_kl(fun(A,B),fun(B,fun(A,B)),F2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,aa(B,fun(A,$o),aTP_Lamp_km(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,aa(B,fun(A,$o),aTP_Lamp_kn(fun(A,B),fun(B,fun(A,$o)),F2),B2)))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_4237_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_4238_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_4239_root__sgn__power,axiom,
    ! [Nb: nat,Y: 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,Y)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_4240_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_4241_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_4242_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_4243_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_4244_split__root,axiom,
    ! [P: fun(real,$o),Nb: nat,Xb: real] :
      ( aa(real,$o,P,aa(real,real,root(Nb),Xb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$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,P,Y4) ) ) ) ) ).

% split_root
tff(fact_4245_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_4246_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_4247_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),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_4248_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_4249_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,L: A,U: A] :
          ( member(A,Ia,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Ia)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ia),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_4250_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_4251_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_4252_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_4253_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_4254_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_4255_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_4256_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_4257_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Xb,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_4258_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,Xb)) = Xb ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_4259_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Xb,Y)) = Xb ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_4260_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,Xb)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_4261_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_4262_inf__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = collect(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3))) ).

% inf_set_def
tff(fact_4263_distinct__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : distinct(product_prod(nat,A),enumerate(A,Nb,Xs)) ).

% distinct_enumerate
tff(fact_4264_shuffles__commutes,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : shuffles(A,Xs,Ys) = shuffles(A,Ys,Xs) ).

% shuffles_commutes
tff(fact_4265_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_4266_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_4267_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_4268_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_4269_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_4270_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_4271_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_4272_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_4273_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,Xb,Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = 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),Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_4274_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4275_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_4276_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_4277_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),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),C2)) = 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),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_4278_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_4279_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),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),C2)) = 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),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_4280_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_4281_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_4282_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C2)
         => ( 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_4283_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_4284_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_ko(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_4285_Fpow__Pow__finite,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A3)),collect(set(A),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_4286_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_4287_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or5935395276787703475ssThan(nat,L,U)) ).

% finite_greaterThanLessThan
tff(fact_4288_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_4289_funpow__0,axiom,
    ! [A: $tType,F2: 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)),F2),Xb) = Xb ).

% funpow_0
tff(fact_4290_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_4291_funpow__mult,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),F2) ).

% funpow_mult
tff(fact_4292_funpow__swap1,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat,Xb: A] : aa(A,A,F2,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),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),F2),aa(A,A,F2,Xb)) ).

% funpow_swap1
tff(fact_4293_empty__in__Fpow,axiom,
    ! [A: $tType,A3: set(A)] : member(set(A),bot_bot(set(A)),finite_Fpow(A,A3)) ).

% empty_in_Fpow
tff(fact_4294_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_4295_Fpow__not__empty,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) != bot_bot(set(set(A))) ).

% Fpow_not_empty
tff(fact_4296_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_4297_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A2) ) ).

% numeral_add_unfold_funpow
tff(fact_4298_Fpow__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),finite_Fpow(A,B3)) ) ).

% Fpow_mono
tff(fact_4299_Fpow__subset__Pow,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),pow2(A,A3)) ).

% Fpow_subset_Pow
tff(fact_4300_Fpow__def,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = collect(set(A),aTP_Lamp_kp(set(A),fun(set(A),$o),A3)) ).

% Fpow_def
tff(fact_4301_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),K) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_4302_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_4303_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_4304_relpowp__fun__conv,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Y: 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,$o))),Nb),P),Xb),Y)
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = Xb )
          & ( aa(nat,A,F7,Nb) = Y )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,F7,I4)),aa(nat,A,F7,aa(nat,nat,suc,I4))) ) ) ) ).

% relpowp_fun_conv
tff(fact_4305_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_4306_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)),insert2(A),Xb),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_4307_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_4308_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_4309_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_4310_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_4311_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_4312_relpowp__0__E,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P),Xb),Y)
     => ( Xb = Y ) ) ).

% relpowp_0_E
tff(fact_4313_relpowp__0__I,axiom,
    ! [A: $tType,P: fun(A,fun(A,$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)),P),Xb),Xb) ).

% relpowp_0_I
tff(fact_4314_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_4315_relpowp__E,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$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,$o))),Nb),P),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M2: nat] :
              ( ( Nb = aa(nat,nat,suc,M2) )
             => ( aa(A,$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))),M2),P),Xb),Y3)
               => ~ aa(A,$o,aa(A,fun(A,$o),P,Y3),Z) ) ) ) ) ).

% relpowp_E
tff(fact_4316_relpowp__E2,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$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,$o))),Nb),P),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M2: nat] :
              ( ( Nb = aa(nat,nat,suc,M2) )
             => ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y3)
               => ~ aa(A,$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))),M2),P),Y3),Z) ) ) ) ) ).

% relpowp_E2
tff(fact_4317_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_4318_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_4319_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)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_4320_sub__num__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,bit1(L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(L))) ) ).

% sub_num_simps(3)
tff(fact_4321_list__update__nonempty,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xb: A] :
      ( ( list_update(A,Xs,K,Xb) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% list_update_nonempty
tff(fact_4322_concat__replicate__trivial,axiom,
    ! [A: $tType,Ia: nat] : concat(A,replicate(list(A),Ia,nil(A))) = nil(A) ).

% concat_replicate_trivial
tff(fact_4323_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_4324_enumerate__simps_I1_J,axiom,
    ! [A: $tType,Nb: nat] : enumerate(A,Nb,nil(A)) = nil(product_prod(nat,A)) ).

% enumerate_simps(1)
tff(fact_4325_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_4326_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_4327_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_4328_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_4329_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_4330_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_4331_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Mb,Nb) ) ).

% diff_numeral_simps(1)
tff(fact_4332_empty__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( nil(A) = replicate(A,Nb,Xb) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_4333_replicate__empty,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( replicate(A,Nb,Xb) = nil(A) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_4334_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A] : groups4207007520872428315er_sum(B,A,F2,A2,nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_4335_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( nil(A) = concat(A,Xss) )
    <=> ! [X2: list(A)] :
          ( member(list(A),X2,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X2 = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_4336_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( concat(A,Xss) = nil(A) )
    <=> ! [X2: list(A)] :
          ( member(list(A),X2,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X2 = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_4337_sub__num__simps_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,bit0(K),bit0(L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(6)
tff(fact_4338_sub__num__simps_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,bit1(K),bit1(L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(9)
tff(fact_4339_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_4340_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V2,W2)),Y) ) ).

% semiring_norm(166)
tff(fact_4341_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W2)),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W2,V2)),Y) ) ).

% semiring_norm(167)
tff(fact_4342_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_4343_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_4344_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(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))) = neg_numeral_sub(A,Nb,Mb) ) ).

% diff_numeral_simps(4)
tff(fact_4345_sub__num__simps_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,bit0(K),bit1(L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(7)
tff(fact_4346_sub__num__simps_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,bit1(K),bit0(L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(8)
tff(fact_4347_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Mb)),one_one(A)) = neg_numeral_sub(A,Mb,one2) ) ).

% diff_numeral_special(2)
tff(fact_4348_diff__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,one2,Nb) ) ).

% diff_numeral_special(1)
tff(fact_4349_sub__num__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,bit1(K),one2) = aa(num,A,numeral_numeral(A),bit0(K)) ) ).

% sub_num_simps(5)
tff(fact_4350_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ( gcd_Gcd(A,A3) = zero_zero(A) )
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).

% Gcd_0_iff
tff(fact_4351_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_4352_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_4353_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_4354_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_4355_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(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_sub(A,one2,Mb) ) ).

% diff_numeral_special(8)
tff(fact_4356_diff__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Nb,one2) ) ).

% diff_numeral_special(7)
tff(fact_4357_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,Mb,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb)) ) ).

% minus_sub_one_diff_one
tff(fact_4358_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_4359_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_4360_shuffles_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Ys),bot_bot(set(list(A)))) ).

% shuffles.simps(1)
tff(fact_4361_shuffles_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] : shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xs),bot_bot(set(list(A)))) ).

% shuffles.simps(2)
tff(fact_4362_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y3: nat] : Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),Y3)) ).

% eq_Abs_Integ
tff(fact_4363_rotate1_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rotate1(A),nil(A)) = nil(A) ).

% rotate1.simps(1)
tff(fact_4364_Gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A),A2: A] :
          ( ! [B4: A] :
              ( member(A,B4,A3)
             => dvd_dvd(A,A2,B4) )
         => dvd_dvd(A,A2,gcd_Gcd(A,A3)) ) ) ).

% Gcd_greatest
tff(fact_4365_dvd__Gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xb: A,A3: set(A)] :
          ( dvd_dvd(A,Xb,gcd_Gcd(A,A3))
        <=> ! [X2: A] :
              ( member(A,X2,A3)
             => dvd_dvd(A,Xb,X2) ) ) ) ).

% dvd_Gcd_iff
tff(fact_4366_dvd__GcdD,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xb: A,A3: set(A),Y: A] :
          ( dvd_dvd(A,Xb,gcd_Gcd(A,A3))
         => ( member(A,Y,A3)
           => dvd_dvd(A,Xb,Y) ) ) ) ).

% dvd_GcdD
tff(fact_4367_Gcd__dvd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => dvd_dvd(A,gcd_Gcd(A,A3),A2) ) ) ).

% Gcd_dvd
tff(fact_4368_remove1_Osimps_I1_J,axiom,
    ! [A: $tType,Xb: A] : remove1(A,Xb,nil(A)) = nil(A) ).

% remove1.simps(1)
tff(fact_4369_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( member(A,one_one(A),A3)
         => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_4370_concat_Osimps_I1_J,axiom,
    ! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).

% concat.simps(1)
tff(fact_4371_list__update__code_I1_J,axiom,
    ! [A: $tType,Ia: nat,Y: A] : list_update(A,nil(A),Ia,Y) = nil(A) ).

% list_update_code(1)
tff(fact_4372_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,Ia: nat,V2: A] : list_update(A,nil(A),Ia,V2) = nil(A) ).

% list_update.simps(1)
tff(fact_4373_Gcd__nat__eq__one,axiom,
    ! [N3: set(nat)] :
      ( member(nat,one_one(nat),N3)
     => ( gcd_Gcd(nat,N3) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_4374_Gcd__dvd__nat,axiom,
    ! [A2: nat,A3: set(nat)] :
      ( member(nat,A2,A3)
     => dvd_dvd(nat,gcd_Gcd(nat,A3),A2) ) ).

% Gcd_dvd_nat
tff(fact_4375_Gcd__greatest__nat,axiom,
    ! [A3: set(nat),A2: nat] :
      ( ! [B4: nat] :
          ( member(nat,B4,A3)
         => dvd_dvd(nat,A2,B4) )
     => dvd_dvd(nat,A2,gcd_Gcd(nat,A3)) ) ).

% Gcd_greatest_nat
tff(fact_4376_int_Oabs__induct,axiom,
    ! [P: fun(int,$o),Xb: int] :
      ( ! [Y3: product_prod(nat,nat)] : aa(int,$o,P,aa(product_prod(nat,nat),int,abs_Integ,Y3))
     => aa(int,$o,P,Xb) ) ).

% int.abs_induct
tff(fact_4377_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_4378_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : distinct(A,nil(A)) ).

% distinct.simps(1)
tff(fact_4379_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_4380_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_4381_replicate__0,axiom,
    ! [A: $tType,Xb: A] : replicate(A,zero_zero(nat),Xb) = nil(A) ).

% replicate_0
tff(fact_4382_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,K,L) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) ) ).

% neg_numeral_class.sub_def
tff(fact_4383_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_4384_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( member(A,A2,A3)
           => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_4385_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,product_case_prod(nat,nat,nat,minus_minus(nat)),Xb) ).

% nat.abs_eq
tff(fact_4386_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_4387_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_4388_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_4389_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_4390_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_4391_zero__int__def,axiom,
    zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))) ).

% zero_int_def
tff(fact_4392_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),product_Pair(nat,nat,Nb),zero_zero(nat))) ).

% int_def
tff(fact_4393_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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ks(nat,fun(nat,product_prod(nat,nat)))),Xb)) ).

% uminus_int.abs_eq
tff(fact_4394_one__int__def,axiom,
    one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_4395_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,product_case_prod(nat,nat,A,aTP_Lamp_kt(nat,fun(nat,A))),Xb) ) ).

% of_int.abs_eq
tff(fact_4396_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),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_kv(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_4397_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),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_kx(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_4398_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)),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)) ).

% plus_int.abs_eq
tff(fact_4399_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lb(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_4400_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow2(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_4401_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,collect(nat,aTP_Lamp_lc(nat,$o))) ) ).

% semiring_char_def
tff(fact_4402_sub__num__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,bit0(L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_4403_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_4404_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_4405_abs__Gcd__eq,axiom,
    ! [K5: set(int)] : aa(int,int,abs_abs(int),gcd_Gcd(int,K5)) = gcd_Gcd(int,K5) ).

% abs_Gcd_eq
tff(fact_4406_iszero__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: num] :
          ( ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),W2)) ) ) ).

% iszero_neg_numeral
tff(fact_4407_dbl__dec__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% dbl_dec_simps(5)
tff(fact_4408_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(bit0(K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_4409_sub__num__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,bit0(K),one2) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% sub_num_simps(4)
tff(fact_4410_Gcd__dvd__int,axiom,
    ! [A2: int,A3: set(int)] :
      ( member(int,A2,A3)
     => dvd_dvd(int,gcd_Gcd(int,A3),A2) ) ).

% Gcd_dvd_int
tff(fact_4411_Gcd__greatest__int,axiom,
    ! [A3: set(int),A2: int] :
      ( ! [B4: int] :
          ( member(int,B4,A3)
         => dvd_dvd(int,A2,B4) )
     => dvd_dvd(int,A2,gcd_Gcd(int,A3)) ) ).

% Gcd_greatest_int
tff(fact_4412_not__iszero__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W2: num] : ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),W2)) ) ).

% not_iszero_numeral
tff(fact_4413_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_4414_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_4415_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_4416_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
        <=> ring_1_iszero(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) ) ) ).

% eq_iff_iszero_diff
tff(fact_4417_Gcd__int__greater__eq__0,axiom,
    ! [K5: set(int)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K5)) ).

% Gcd_int_greater_eq_0
tff(fact_4418_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xb)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_4419_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_4420_not__iszero__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),one2)) ) ).

% not_iszero_Numeral1
tff(fact_4421_not__iszero__neg__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_iszero_neg_1
tff(fact_4422_eq__numeral__iff__iszero_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Xb,Y)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_4423_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa(num,nat,numeral_numeral(nat),bit0(Nb)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(Nb))) ).

% eval_nat_numeral(2)
tff(fact_4424_one__plus__BitM,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(Nb)) = bit0(Nb) ).

% one_plus_BitM
tff(fact_4425_BitM__plus__one,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(Nb)),one2) = bit0(Nb) ).

% BitM_plus_one
tff(fact_4426_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_4427_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xb)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_4428_not__iszero__neg__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) ) ).

% not_iszero_neg_Numeral1
tff(fact_4429_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_4430_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_4431_eq__numeral__iff__iszero_I4_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Y,Xb)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_4432_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),bitM(Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),bit0(Nb))),one_one(A)) ) ).

% numeral_BitM
tff(fact_4433_sub__BitM__One__eq,axiom,
    ! [Nb: num] : neg_numeral_sub(int,bitM(Nb),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),neg_numeral_sub(int,Nb,one2)) ).

% sub_BitM_One_eq
tff(fact_4434_eq__numeral__iff__iszero_I5_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = one_one(A) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Xb,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_4435_eq__numeral__iff__iszero_I6_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_4436_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_4437_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),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_kx(nat,fun(nat,fun(product_prod(nat,nat),$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_4438_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),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_kv(nat,fun(nat,fun(product_prod(nat,nat),$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_4439_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P3: fun(A,B),Ia: A] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( groups1962203154675924110t_prod(A,B,P3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ia),I5)) = $ite(member(A,Ia,I5),groups1962203154675924110t_prod(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P3,Ia)),groups1962203154675924110t_prod(A,B,P3,I5))) ) ) ) ).

% prod.insert'
tff(fact_4440_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A)] : groups1962203154675924110t_prod(B,A,P3,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_4441_prod_Oeq__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1962203154675924110t_prod(A,B,P3,I5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),P3),I5) ) ) ) ).

% prod.eq_sum
tff(fact_4442_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ld(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_4443_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G,X3) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,S2) = groups1962203154675924110t_prod(A,B,G,T3) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_4444_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G,X3) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,T3) = groups1962203154675924110t_prod(A,B,G,S2) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_4445_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),T3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,H,I2) = one_one(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S2)
                 => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
             => ( groups1962203154675924110t_prod(A,B,G,S2) = groups1962203154675924110t_prod(A,B,H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_4446_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),T3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G,X3) = one_one(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S2)
                 => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
             => ( groups1962203154675924110t_prod(A,B,G,T3) = groups1962203154675924110t_prod(A,B,H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_4447_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ld(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_4448_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1962203154675924110t_prod(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_le(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_le(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),one_one(A)) ) ).

% prod.G_def
tff(fact_4449_nat_Orep__eq,axiom,
    ! [Xb: int] : aa(int,nat,nat2,Xb) = aa(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_4450_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,product_case_prod(nat,nat,A,aTP_Lamp_kt(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,Xb)) ) ).

% of_int.rep_eq
tff(fact_4451_uminus__int__def,axiom,
    uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ks(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_4452_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) = remove1(A,Xb,aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_4453_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_4454_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xb: nat,Y: nat] :
      aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_lf(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,Xb,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4455_image__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),Xb: B,A3: set(B)] :
      ( ( B2 = aa(B,A,F2,Xb) )
     => ( member(B,Xb,A3)
       => member(A,B2,aa(set(B),set(A),image2(B,A,F2),A3)) ) ) ).

% image_eqI
tff(fact_4456_image__ident,axiom,
    ! [A: $tType,Y5: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_lg(A,A)),Y5) = Y5 ).

% image_ident
tff(fact_4457_image__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_4458_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F2),A3) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_4459_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),A3) = bot_bot(set(A)) )
    <=> ( A3 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_4460_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_4461_image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: B,B3: set(B)] : aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,F2,A2)),aa(set(B),set(A),image2(B,A,F2),B3)) ).

% image_insert
tff(fact_4462_insert__image,axiom,
    ! [B: $tType,A: $tType,Xb: A,A3: set(A),F2: fun(A,B)] :
      ( member(A,Xb,A3)
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,Xb)),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),set(B),image2(A,B,F2),A3) ) ) ).

% insert_image
tff(fact_4463_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_4464_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_4465_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,Ia: A,J: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or1337092689740270186AtMost(A,Ia,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost
tff(fact_4466_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D2),A2)) ) ).

% image_diff_atLeastAtMost
tff(fact_4467_image__uminus__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or1337092689740270186AtMost(A,Xb,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_atLeastAtMost
tff(fact_4468_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,Ia: A,J: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or7035219750837199246ssThan(A,Ia,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan
tff(fact_4469_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C2)),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),C2),A2)) ) ).

% image_add_atMost
tff(fact_4470_image__uminus__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or5935395276787703475ssThan(A,Xb,Y)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_greaterThanLessThan
tff(fact_4471_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_4472_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_4473_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(list(A),set(A),set2(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) = A3 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_4474_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(list(A),nat,size_size(list(A)),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) = aa(set(A),nat,finite_card(A),A3) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_4475_image__Suc__atLeastAtMost,axiom,
    ! [Ia: nat,J: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,Ia,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Ia),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_4476_image__Suc__atLeastLessThan,axiom,
    ! [Ia: nat,J: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,Ia,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Ia),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_4477_ccSUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lh(B,A)),A3)) = bot_bot(A) ) ).

% ccSUP_bot
tff(fact_4478_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_li(B,A)),A3)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_4479_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A3)) = bot_bot(A) )
        <=> ! [X2: B] :
              ( member(B,X2,A3)
             => ( aa(B,A,B3,X2) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_4480_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: fun(B,A),A3: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B3),A3)) )
        <=> ! [X2: B] :
              ( member(B,X2,A3)
             => ( aa(B,A,B3,X2) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_4481_cSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),C2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lj(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).

% cSUP_const
tff(fact_4482_ccSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lk(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% ccSUP_const
tff(fact_4483_SUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ll(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% SUP_const
tff(fact_4484_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,Ia: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_lm(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,Ia,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_4485_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),C2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lj(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).

% cINF_const
tff(fact_4486_ccINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_lk(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% ccINF_const
tff(fact_4487_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ll(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% INF_const
tff(fact_4488_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ln(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_4489_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,Ia: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_lm(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,Ia,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Ia),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_4490_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = nil(A) )
          <=> ( A3 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4491_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)) = bot_bot(A) )
        <=> ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X2)
             => ? [Xa3: B] :
                  ( member(B,Xa3,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Xa3)),X2) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_4492_ccSUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).

% ccSUP_empty
tff(fact_4493_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4494_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_lo(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,divide_divide(A,A2,D2),divide_divide(A,B2,D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4495_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,F2),A3))
       => ? [X3: A] :
            ( member(A,X3,A3)
            & ~ aa(set(A),$o,finite_finite2(A),collect(A,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_lp(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A3),F2),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_4496_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_lq(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F2),A3),P)) = aa(set(B),set(A),image2(B,A,F2),collect(B,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_lr(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),A3),P))) ).

% Compr_image_eq
tff(fact_4497_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image2(B,A,F2),aa(set(C),set(B),image2(C,B,G),A3)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_ls(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A3) ).

% image_image
tff(fact_4498_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),A3: set(B)] :
      ( member(A,B2,aa(set(B),set(A),image2(B,A,F2),A3))
     => ~ ! [X3: B] :
            ( ( B2 = aa(B,A,F2,X3) )
           => ~ member(B,X3,A3) ) ) ).

% imageE
tff(fact_4499_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,Xb: A,A3: set(A),B2: B,F2: fun(A,B)] :
      ( member(A,Xb,A3)
     => ( ( B2 = aa(A,B,F2,Xb) )
       => member(B,B2,aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ).

% rev_image_eqI
tff(fact_4500_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A3))
         => aa(A,$o,P,X3) )
     => ! [X: B] :
          ( member(B,X,A3)
         => aa(A,$o,P,aa(B,A,F2,X)) ) ) ).

% ball_imageD
tff(fact_4501_image__cong,axiom,
    ! [B: $tType,A: $tType,M4: set(A),N3: set(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( M4 = N3 )
     => ( ! [X3: A] :
            ( member(A,X3,N3)
           => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) )
       => ( aa(set(A),set(B),image2(A,B,F2),M4) = aa(set(A),set(B),image2(A,B,G),N3) ) ) ) ).

% image_cong
tff(fact_4502_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(set(B),set(A),image2(B,A,F2),A3))
          & aa(A,$o,P,X) )
     => ? [X3: B] :
          ( member(B,X3,A3)
          & aa(A,$o,P,aa(B,A,F2,X3)) ) ) ).

% bex_imageD
tff(fact_4503_image__iff,axiom,
    ! [A: $tType,B: $tType,Z: A,F2: fun(B,A),A3: set(B)] :
      ( member(A,Z,aa(set(B),set(A),image2(B,A,F2),A3))
    <=> ? [X2: B] :
          ( member(B,X2,A3)
          & ( Z = aa(B,A,F2,X2) ) ) ) ).

% image_iff
tff(fact_4504_imageI,axiom,
    ! [B: $tType,A: $tType,Xb: A,A3: set(A),F2: fun(A,B)] :
      ( member(A,Xb,A3)
     => member(B,aa(A,B,F2,Xb),aa(set(A),set(B),image2(A,B,F2),A3)) ) ).

% imageI
tff(fact_4505_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),A3)
          & ( B3 = aa(set(B),set(A),image2(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_4506_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B3)
    <=> ! [X2: B] :
          ( member(B,X2,A3)
         => member(A,aa(B,A,F2,X2),B3) ) ) ).

% image_subset_iff
tff(fact_4507_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3))
     => ~ ! [C8: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C8),A3)
           => ( B3 != aa(set(B),set(A),image2(B,A,F2),C8) ) ) ) ).

% subset_imageE
tff(fact_4508_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B3: set(B)] :
      ( ! [X3: A] :
          ( member(A,X3,A3)
         => member(B,aa(A,B,F2,X3),B3) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3) ) ).

% image_subsetI
tff(fact_4509_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3)) ) ).

% image_mono
tff(fact_4510_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ! [B11: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F2),A3))
         => aa(set(A),$o,P,B11) )
    <=> ! [B11: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A3)
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),B11)) ) ) ).

% all_subset_image
tff(fact_4511_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),A3) = B3 )
     => ( aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow2(B,A3)) = pow2(A,B3) ) ) ).

% image_Pow_surj
tff(fact_4512_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(A,B),B3: set(B)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
         => member(B,aa(A,B,F2,X3),B3) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),collect(A,P))),B3) ) ).

% image_Collect_subsetI
tff(fact_4513_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B3)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow2(B,A3))),pow2(A,B3)) ) ).

% image_Pow_mono
tff(fact_4514_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(set(A),list(A),linord4507533701916653071of_set(A),B3) )
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( A3 = B3 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_4515_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_lt(fun(B,A),fun(B,set(A)),F2)),A3)) = aa(set(B),set(A),image2(B,A,F2),A3) ).

% UNION_singleton_eq_range
tff(fact_4516_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : distinct(A,aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_4517_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B3)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),finite_Fpow(B,A3))),finite_Fpow(A,B3)) ) ).

% image_Fpow_mono
tff(fact_4518_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B3: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => ? [X: B] :
                  ( member(B,X,B3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,I2)),aa(B,C,G,X)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B3)
               => ? [X: A] :
                    ( member(A,X,A3)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,J2)),aa(A,C,F2,X)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,F2),A3)) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B3)) ) ) ) ) ).

% SUP_eq
tff(fact_4519_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B3: set(B),G: fun(B,C),F2: fun(A,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => ? [X: B] :
                  ( member(B,X,B3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,X)),aa(A,C,F2,I2)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B3)
               => ? [X: A] :
                    ( member(A,X,A3)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X)),aa(B,C,G,J2)) ) )
           => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,F2),A3)) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),B3)) ) ) ) ) ).

% INF_eq
tff(fact_4520_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ member(nat,zero_zero(nat),aa(set(nat),set(nat),image2(nat,nat,suc),A3)) ).

% zero_notin_Suc_image
tff(fact_4521_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ! [B11: set(A)] :
          ( ( aa(set(A),$o,finite_finite2(A),B11)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F2),A3)) )
         => aa(set(A),$o,P,B11) )
    <=> ! [B11: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),B11)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A3) )
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),B11)) ) ) ).

% all_finite_subset_image
tff(fact_4522_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ? [B11: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B11)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F2),A3))
          & aa(set(A),$o,P,B11) )
    <=> ? [B11: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),B11)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A3)
          & aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),B11)) ) ) ).

% ex_finite_subset_image
tff(fact_4523_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3))
       => ? [C8: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C8),A3)
            & aa(set(B),$o,finite_finite2(B),C8)
            & ( B3 = aa(set(B),set(A),image2(B,A,F2),C8) ) ) ) ) ).

% finite_subset_image
tff(fact_4524_finite__surj,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F2),A3))
       => aa(set(B),$o,finite_finite2(B),B3) ) ) ).

% finite_surj
tff(fact_4525_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_4526_SUP__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),Xb: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => ( aa(A,B,F2,I2) = Xb ) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),I5)) = Xb ) ) ) ) ).

% SUP_eq_const
tff(fact_4527_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_4528_INF__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),Xb: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => ( aa(A,B,F2,I2) = Xb ) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I5)) = Xb ) ) ) ) ).

% INF_eq_const
tff(fact_4529_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_4530_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),aa(set(B),set(A),image2(B,A,F2),B3))) ).

% image_Int_subset
tff(fact_4531_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : aa(set(A),$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)),aa(set(B),set(A),image2(B,A,F2),A3)),aa(set(B),set(A),image2(B,A,F2),B3))),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3))) ).

% image_diff_subset
tff(fact_4532_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_4533_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B),Xb: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),Xb) )
         => ( ! [Y3: B] :
                ( ! [I3: A] :
                    ( member(A,I3,A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),Y3) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),Y3) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3)) = Xb ) ) ) ) ).

% SUP_eqI
tff(fact_4534_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B3: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [N: A] :
              ( member(A,N,A3)
             => ? [X: B] :
                  ( member(B,X,B3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,N)),aa(B,C,G,X)) ) )
         => aa(C,$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,F2),A3))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B3))) ) ) ).

% SUP_mono
tff(fact_4535_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),U) ) ) ).

% SUP_least
tff(fact_4536_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A3: set(A)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),A3))) ) ) ).

% SUP_mono'
tff(fact_4537_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia: A,A3: set(A),F2: fun(A,B)] :
          ( member(A,Ia,A3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ia)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ).

% SUP_upper
tff(fact_4538_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),U)
        <=> ! [X2: B] :
              ( member(B,X2,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X2)),U) ) ) ) ).

% SUP_le_iff
tff(fact_4539_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia: A,A3: set(A),U: B,F2: fun(A,B)] :
          ( member(A,Ia,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,Ia))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ).

% SUP_upper2
tff(fact_4540_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))
        <=> ? [X2: B] :
              ( member(B,X2,A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,X2)) ) ) ) ).

% less_SUP_iff
tff(fact_4541_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,Ia: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),Y)
         => ( member(B,Ia,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Ia)),Y) ) ) ) ).

% SUP_lessD
tff(fact_4542_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),Xb: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),aa(A,B,F2,I2)) )
         => ( ! [Y3: B] :
                ( ! [I3: A] :
                    ( member(A,I3,A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),aa(A,B,F2,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),Xb) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3)) = Xb ) ) ) ) ).

% INF_eqI
tff(fact_4543_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B3: set(A),A3: set(B),F2: fun(B,C),G: fun(A,C)] :
          ( ! [M2: A] :
              ( member(A,M2,B3)
             => ? [X: B] :
                  ( member(B,X,A3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F2,X)),aa(A,C,G,M2)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F2),A3))),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,G),B3))) ) ) ).

% INF_mono
tff(fact_4544_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia: A,A3: set(A),F2: fun(A,B)] :
          ( member(A,Ia,A3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,Ia)) ) ) ).

% INF_lower
tff(fact_4545_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A3: set(A)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),A3))) ) ) ).

% INF_mono'
tff(fact_4546_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Ia: A,A3: set(A),F2: fun(A,B),U: B] :
          ( member(A,Ia,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ia)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),U) ) ) ) ).

% INF_lower2
tff(fact_4547_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)))
        <=> ! [X2: B] :
              ( member(B,X2,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F2,X2)) ) ) ) ).

% le_INF_iff
tff(fact_4548_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),U: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ).

% INF_greatest
tff(fact_4549_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),A2)
        <=> ? [X2: B] :
              ( member(B,X2,A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),A2) ) ) ) ).

% INF_less_iff
tff(fact_4550_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F2: fun(B,A),A3: set(B),Ia: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)))
         => ( member(B,Ia,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,Ia)) ) ) ) ).

% less_INF_D
tff(fact_4551_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
    <=> ? [N2: nat,F7: fun(nat,A)] : A3 = aa(set(nat),set(A),image2(nat,A,F7),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),N2))) ) ).

% finite_conv_nat_seg_image
tff(fact_4552_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F2: fun(nat,A),Nb: nat] :
      ( ( A3 = aa(set(nat),set(A),image2(nat,A,F2),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb))) )
     => aa(set(A),$o,finite_finite2(A),A3) ) ).

% nat_seg_image_imp_finite
tff(fact_4553_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A3: set(B)] :
      aa(set(B),set(A),image2(B,A,aTP_Lamp_lu(A,fun(B,A),C2)),A3) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C2),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_4554_image__constant,axiom,
    ! [A: $tType,B: $tType,Xb: A,A3: set(A),C2: B] :
      ( member(A,Xb,A3)
     => ( aa(set(A),set(B),image2(A,B,aTP_Lamp_lv(B,fun(A,B),C2)),A3) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_4555_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),H: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$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_lx(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H),G)),aa(set(A),set(C),image2(A,C,G),S2)) ) ) ) ).

% sum.image_gen
tff(fact_4556_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),H: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$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_ly(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H),G)),aa(set(A),set(C),image2(A,C,G),S2)) ) ) ) ).

% prod.image_gen
tff(fact_4557_Gcd__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_Gcd(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => dvd_dvd(B,aa(A,B,F2,X3),aa(A,B,G,X3)) )
         => dvd_dvd(B,gcd_Gcd(B,aa(set(A),set(B),image2(A,B,F2),A3)),gcd_Gcd(B,aa(set(A),set(B),image2(A,B,G),A3))) ) ) ).

% Gcd_mono
tff(fact_4558_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),Xb: A] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ! [Y3: A] :
            ( member(A,Y3,A3)
           => ( aa(A,B,F2,Y3) = aa(A,B,F2,Xb) ) )
       => ( the_elem(B,aa(set(A),set(B),image2(A,B,F2),A3)) = aa(A,B,F2,Xb) ) ) ) ).

% the_elem_image_unique
tff(fact_4559_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)))
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xb)
             => ? [X2: B] :
                  ( member(B,X2,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),aa(B,A,F2,X2)) ) ) ) ) ).

% le_SUP_iff
tff(fact_4560_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),Xb)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y4)
             => ? [X2: B] :
                  ( member(B,X2,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),Y4) ) ) ) ) ).

% INF_le_iff
tff(fact_4561_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),M4: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),M4) )
           => aa(B,$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,F2),A3))),M4) ) ) ) ).

% cSUP_least
tff(fact_4562_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),C2: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(A,B,F2,I2)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),I5)) = C2 )
            <=> ! [X2: A] :
                  ( member(A,X2,I5)
                 => ( aa(A,B,F2,X2) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_4563_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),Mb: B,F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F2,X3)) )
           => aa(B,$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,F2),A3))) ) ) ) ).

% cINF_greatest
tff(fact_4564_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),C2: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),C2) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I5)) = C2 )
            <=> ! [X2: A] :
                  ( member(A,X2,I5)
                 => ( aa(A,B,F2,X2) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4565_card__image__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(A),nat,finite_card(A),A3)) ) ).

% card_image_le
tff(fact_4566_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ).

% SUP_subset_mono
tff(fact_4567_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => ( ! [X3: A] :
                ( member(A,X3,B3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ).

% INF_superset_mono
tff(fact_4568_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_4569_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A3: set(B)] :
          aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lz(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(A),C2) ) ).

% SUP_constant
tff(fact_4570_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S2: set(A),T3: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S2)),T3)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mb(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G),H)),T3) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),H),S2) ) ) ) ) ) ).

% sum.group
tff(fact_4571_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),Xb: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mc(B,fun(fun(A,B),fun(A,B)),Xb),F2)),I5)) = 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,F2),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_4572_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),Xb: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_md(fun(A,B),fun(B,fun(A,B)),F2),Xb)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I5))),Xb) ) ) ) ).

% INF_inf_const2
tff(fact_4573_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S2: set(A),T3: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S2)),T3)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_me(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G),H)),T3) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),H),S2) ) ) ) ) ) ).

% prod.group
tff(fact_4574_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ).

% INF_le_SUP
tff(fact_4575_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F2),A3))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B3)),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% surj_card_le
tff(fact_4576_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(set(A),set(A),image2(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),Xb),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_4577_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_4578_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_4579_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)),insert2(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_4580_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)),insert2(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_4581_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)),insert2(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_4582_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)),insert2(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_4583_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Xb)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4584_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          aa(set(A),set(A),image2(A,A,aTP_Lamp_mf(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4585_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_mg(A,fun(A,fun(A,A)),Mb),C2)),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)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2)),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)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4586_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_mh(A,fun(A,fun(A,A)),Mb),C2)),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)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2)),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)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4587_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_mi(A,fun(A,fun(A,A)),Mb),C2)),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)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,Mb)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,Mb)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Mb)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4588_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_mj(A,fun(A,fun(A,A)),Mb),C2)),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)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,Mb)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,Mb)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Mb)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4589_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S2: set(A),R: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( aa(set(A),$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,G),S2)),R)
             => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_mk(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S2),G),F2)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_4590_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,Ia: 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,Ia)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,Ia,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_4591_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_mm(A,fun(nat,A),B3)),collect(nat,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ).

% INF_nat_binary
tff(fact_4592_times__int__def,axiom,
    times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_4593_minus__int__def,axiom,
    minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lb(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_4594_plus__int__def,axiom,
    plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_kz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_4595_Gcd__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,aa(set(int),set(int),image2(int,int,abs_abs(int)),K5)) = gcd_Gcd(int,K5) ).

% Gcd_abs_eq
tff(fact_4596_finite__UN,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3)))
      <=> ! [X2: A] :
            ( member(A,X2,A3)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,X2)) ) ) ) ).

% finite_UN
tff(fact_4597_UN__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_mn(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),C2) ).

% UN_constant
tff(fact_4598_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [A4: A] :
            ( member(A,A4,A3)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,A4)) )
       => aa(set(B),$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),B3),A3))) ) ) ).

% finite_UN_I
tff(fact_4599_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( ? [X: A] :
          ( member(A,X,I5)
          & aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,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),A3),I5))) ) ).

% finite_INT
tff(fact_4600_Gcd__int__eq,axiom,
    ! [N3: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),N3)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N3)) ).

% Gcd_int_eq
tff(fact_4601_UN__singleton,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_mo(A,set(A))),A3)) = A3 ).

% UN_singleton
tff(fact_4602_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B3: 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_mp(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B3)),C5)) = $ite(C5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C5)))) ).

% UN_simps(1)
tff(fact_4603_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_4604_Gcd__nat__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aTP_Lamp_mq(int,nat)),K5)) = aa(int,nat,nat2,gcd_Gcd(int,K5)) ).

% Gcd_nat_abs_eq
tff(fact_4605_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F4: fun(A,filter(B))] :
      ( ! [X6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),B3)
         => ( aa(set(A),$o,finite_finite2(A),X6)
           => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),X6)) != bot_bot(filter(B)) ) ) )
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B3)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_4606_finite__int__iff__bounded,axiom,
    ! [S2: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S2)
    <=> ? [K2: int] : aa(set(int),$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),K2)) ) ).

% finite_int_iff_bounded
tff(fact_4607_finite__int__iff__bounded__le,axiom,
    ! [S2: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S2)
    <=> ? [K2: int] : aa(set(int),$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),K2)) ) ).

% finite_int_iff_bounded_le
tff(fact_4608_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A3: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A3)) = bot_bot(set(A)) )
    <=> ! [X2: B] :
          ( member(B,X2,A3)
         => ( aa(B,set(A),B3,X2) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_4609_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A)),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),A3)) )
    <=> ! [X2: B] :
          ( member(B,X2,A3)
         => ( aa(B,set(A),B3,X2) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_4610_UN__empty,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_4611_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_mr(B,set(A))),A3)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_4612_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I5: set(B),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),I5))),B3)
    <=> ! [X2: B] :
          ( member(B,X2,I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),A3,X2)),B3) ) ) ).

% UN_subset_iff
tff(fact_4613_UN__upper,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B3: fun(A,set(B))] :
      ( member(A,A2,A3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B3,A2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3))) ) ).

% UN_upper
tff(fact_4614_UN__least,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(A,set(B)),C5: set(B)] :
      ( ! [X3: A] :
          ( member(A,X3,A3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B3,X3)),C5) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3))),C5) ) ).

% UN_least
tff(fact_4615_UN__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( ! [X3: A] :
            ( member(A,X3,A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,X3)),aa(A,set(B),G,X3)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),A3))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),B3))) ) ) ).

% UN_mono
tff(fact_4616_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A3: fun(B,set(A)),I5: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),I5)))
    <=> ! [X2: B] :
          ( member(B,X2,I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(B,set(A),A3,X2)) ) ) ).

% INT_subset_iff
tff(fact_4617_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( ! [X3: A] :
            ( member(A,X3,A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,X3)),aa(A,set(B),G,X3)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),B3))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),A3))) ) ) ).

% INT_anti_mono
tff(fact_4618_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C5: set(B),B3: fun(A,set(B))] :
      ( ! [X3: A] :
          ( member(A,X3,A3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C5),aa(A,set(B),B3,X3)) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C5),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3))) ) ).

% INT_greatest
tff(fact_4619_INT__lower,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B3: fun(A,set(B))] :
      ( member(A,A2,A3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3))),aa(A,set(B),B3,A2)) ) ).

% INT_lower
tff(fact_4620_in__image__insert__iff,axiom,
    ! [A: $tType,B3: set(set(A)),Xb: A,A3: set(A)] :
      ( ! [C8: set(A)] :
          ( member(set(A),C8,B3)
         => ~ member(A,Xb,C8) )
     => ( member(set(A),A3,aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb)),B3))
      <=> ( member(A,Xb,A3)
          & member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))),B3) ) ) ) ).

% in_image_insert_iff
tff(fact_4621_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_4622_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B3: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C5))) = $ite(C5 = bot_bot(set(B)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(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_mp(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B3)),C5))) ).

% UN_extend_simps(1)
tff(fact_4623_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_4624_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B3: 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),A3),C5))),B3) = $ite(C5 = bot_bot(set(B)),B3,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_ms(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C5))) ).

% INT_extend_simps(1)
tff(fact_4625_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C5))) = $ite(C5 = bot_bot(set(B)),A3,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_mt(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C5))) ).

% INT_extend_simps(2)
tff(fact_4626_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,A3: set(A),B12: set(set(A))] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)) = $ite(B12 = bot_bot(set(set(A))),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3)),B12))) ).

% Int_Inter_eq(1)
tff(fact_4627_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B12: set(set(A)),A3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12)),A3) = $ite(B12 = bot_bot(set(set(A))),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aTP_Lamp_mu(set(A),fun(set(A),set(A)),A3)),B12))) ).

% Int_Inter_eq(2)
tff(fact_4628_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C5))) = $ite(C5 = bot_bot(set(B)),A3,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_mv(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C5))) ).

% INT_extend_simps(4)
tff(fact_4629_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),K: 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_mw(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M4),K)),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,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ).

% UN_le_add_shift_strict
tff(fact_4630_UN__le__add__shift,axiom,
    ! [A: $tType,M4: fun(nat,set(A)),K: 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_mw(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M4),K)),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,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ).

% UN_le_add_shift
tff(fact_4631_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_4632_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_4633_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image2(int,int,aTP_Lamp_mx(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_4634_sum_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(A),A3: fun(A,set(B)),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [X3: A] :
                ( member(A,X3,I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X3)) )
           => ( ! [X3: A] :
                  ( member(A,X3,I5)
                 => ! [Xa2: A] :
                      ( member(A,Xa2,I5)
                     => ( ( X3 != Xa2 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X3)),aa(A,set(B),A3,Xa2)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_my(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_4635_prod_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(A),A3: fun(A,set(B)),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [X3: A] :
                ( member(A,X3,I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X3)) )
           => ( ! [X3: A] :
                  ( member(A,X3,I5)
                 => ! [Xa2: A] :
                      ( member(A,Xa2,I5)
                     => ( ( X3 != Xa2 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X3)),aa(A,set(B),A3,Xa2)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_mz(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_4636_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_na(fun(A,set(B)),fun(A,nat),A3)),I5)) ) ).

% card_UN_le
tff(fact_4637_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_4638_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [X3: A] :
            ( member(A,X3,I5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A3,X3)) )
       => ( ! [X3: A] :
              ( member(A,X3,I5)
             => ! [Xa2: A] :
                  ( member(A,Xa2,I5)
                 => ( ( X3 != Xa2 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X3)),aa(A,set(B),A3,Xa2)) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_na(fun(A,set(B)),fun(A,nat),A3)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_4639_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),G: fun(C,set(B)),Xb: C,X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),aa(C,set(B),G,Xb)))),X5)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),G,Xb)),collect(B,aa(set(A),fun(B,$o),aTP_Lamp_nb(fun(B,set(A)),fun(set(A),fun(B,$o)),F2),X5))) ) ).

% UN_image_subset
tff(fact_4640_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S2: set(A)] :
      ( ! [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F2,I2)),S2)
     => ( aa(set(A),$o,finite_finite2(A),S2)
       => ( ? [N7: nat] :
              ( ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N7)
                 => ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N7)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F2,M2)),aa(nat,set(A),F2,N)) ) ) )
              & ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
                 => ( aa(nat,set(A),F2,N7) = aa(nat,set(A),F2,N) ) ) )
         => ( aa(nat,set(A),F2,aa(set(A),nat,finite_card(A),S2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),F2),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_4641_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_4642_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_4643_UNIV__I,axiom,
    ! [A: $tType,Xb: A] : member(A,Xb,top_top(set(A))) ).

% UNIV_I
tff(fact_4644_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_4645_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_4646_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_4647_Int__UNIV,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = top_top(set(A)) )
    <=> ( ( A3 = top_top(set(A)) )
        & ( B3 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_4648_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_4649_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_4650_Pow__UNIV,axiom,
    ! [A: $tType] : pow2(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_4651_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_4652_remdups__eq__nil__iff,axiom,
    ! [A: $tType,Xb: list(A)] :
      ( ( remdups(A,Xb) = nil(A) )
    <=> ( Xb = nil(A) ) ) ).

% remdups_eq_nil_iff
tff(fact_4653_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_4654_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_4655_remdups__id__iff__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( remdups(A,Xs) = Xs )
    <=> distinct(A,Xs) ) ).

% remdups_id_iff_distinct
tff(fact_4656_distinct__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : distinct(A,remdups(A,Xs)) ).

% distinct_remdups
tff(fact_4657_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      collect(A,aTP_Lamp_nc($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_4658_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,P))
     => ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_be(fun(A,$o),fun(A,$o),P)))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_4659_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_4660_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_4661_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A3) = top_top(A) )
        <=> ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),top_top(A))
             => ? [Xa3: A] :
                  ( member(A,Xa3,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Xa3) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_4662_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_4663_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_4664_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_4665_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_4666_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_4667_Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_4668_surj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( ( aa(set(A),set(A),image2(A,A,F2),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image2(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_4669_finite__compl,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),uminus_uminus(set(A)),A3))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_compl
tff(fact_4670_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_4671_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_4672_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3)) = top_top(A) )
        <=> ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),top_top(A))
             => ? [Xa3: B] :
                  ( member(B,Xa3,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),aa(B,A,F2,Xa3)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_4673_range__constant,axiom,
    ! [B: $tType,A: $tType,Xb: A] : aa(set(B),set(A),image2(B,A,aTP_Lamp_lu(A,fun(B,A),Xb)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) ).

% range_constant
tff(fact_4674_ccINF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).

% ccINF_empty
tff(fact_4675_INT__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A3: 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_mn(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(set(A)),C2) ).

% INT_constant
tff(fact_4676_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_4677_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: 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_mt(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),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)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C5)))) ).

% INT_simps(2)
tff(fact_4678_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B3: 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_ms(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),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),A3),C5))),B3)) ).

% INT_simps(1)
tff(fact_4679_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B3: 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_nd(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),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),A3),C5))),B3)) ).

% INT_simps(3)
tff(fact_4680_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: 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_mv(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),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)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C5)))) ).

% INT_simps(4)
tff(fact_4681_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),Xb: B] :
      ( ( B2 = aa(B,A,F2,Xb) )
     => member(A,B2,aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ) ).

% range_eqI
tff(fact_4682_rangeI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xb: B] : member(A,aa(B,A,F2,Xb),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ).

% rangeI
tff(fact_4683_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F4: fun(A,filter(B))] :
      ( ! [I2: A] :
          ( member(A,I2,I5)
         => ! [J2: A] :
              ( member(A,J2,I5)
             => ? [X: A] :
                  ( member(A,X,I5)
                  & aa(filter(B),$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,I2)),aa(A,filter(B),F4,J2))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),I5)) = bot_bot(filter(B)) )
      <=> ? [X2: A] :
            ( member(A,X2,I5)
            & ( aa(A,filter(B),F4,X2) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_4684_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A)] :
      ( member(A,B2,aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
     => ~ ! [X3: B] : B2 != aa(B,A,F2,X3) ) ).

% rangeE
tff(fact_4685_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ne(fun(C,A),fun(fun(B,C),fun(B,A)),F2),G)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F2),aa(set(B),set(C),image2(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_4686_remdups_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups(A,nil(A)) = nil(A) ).

% remdups.simps(1)
tff(fact_4687_Int__UNIV__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B3) = B3 ).

% Int_UNIV_left
tff(fact_4688_Int__UNIV__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),top_top(set(A))) = A3 ).

% Int_UNIV_right
tff(fact_4689_distinct__remdups__id,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups(A,Xs) = Xs ) ) ).

% distinct_remdups_id
tff(fact_4690_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_nf(A,$o)) ).

% UNIV_def
tff(fact_4691_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_4692_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X3: A] : member(A,X3,top_top(set(A))) ).

% UNIV_witness
tff(fact_4693_UNIV__eq__I,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X3: A] : member(A,X3,A3)
     => ( top_top(set(A)) = A3 ) ) ).

% UNIV_eq_I
tff(fact_4694_remdups__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : remdups(A,remdups(A,Xs)) = remdups(A,Xs) ).

% remdups_remdups
tff(fact_4695_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_4696_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_4697_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_4698_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_4699_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_4700_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_4701_ex__new__if__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),A3)
       => ? [A4: A] : ~ member(A,A4,A3) ) ) ).

% ex_new_if_finite
tff(fact_4702_finite__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_UNIV
tff(fact_4703_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_4704_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_4705_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_4706_infinite__UNIV__nat,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% infinite_UNIV_nat
tff(fact_4707_nat__not__finite,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% nat_not_finite
tff(fact_4708_subset__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),top_top(set(A))) ).

% subset_UNIV
tff(fact_4709_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_4710_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_4711_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_4712_Inf__filter__not__bot,axiom,
    ! [A: $tType,B3: set(filter(A))] :
      ( ! [X6: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X6),B3)
         => ( aa(set(filter(A)),$o,finite_finite2(filter(A)),X6)
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X6) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_4713_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_4714_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [Xb: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,Xb,Y) = top_top(set(A)) )
        <=> ( ( Xb = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_4715_insert__UNIV,axiom,
    ! [A: $tType,Xb: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_4716_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_4717_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_4718_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_4719_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B3: set(A),Ia: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))),B3)
     => member(A,aa(B,A,F2,Ia),B3) ) ).

% range_subsetD
tff(fact_4720_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)),insert2(A),Xb),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_4721_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_4722_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
       => ( A3 = top_top(set(A)) ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_4723_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_4724_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_4725_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_4726_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_4727_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A3) ).

% Compl_eq_Diff_UNIV
tff(fact_4728_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_4729_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F2: fun(A,C)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,G),top_top(set(B))))
     => aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_ng(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_4730_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_4731_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: A,Xb: B] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))) )
     => ( aa(B,A,F2,Xb) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4732_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A3: set(B)] :
          aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_lz(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(A),C2) ) ).

% INF_constant
tff(fact_4733_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_4734_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),uminus_uminus(set(B)),A3))) ) ).

% surj_Compl_image_subset
tff(fact_4735_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_4736_INT__empty,axiom,
    ! [B: $tType,A: $tType,B3: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_4737_UN__UN__finite__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aTP_Lamp_nh(fun(nat,set(A)),fun(nat,set(A)),A3)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_4738_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_4739_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_4740_conj__subset__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,P))
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,Q)) ) ) ).

% conj_subset_def
tff(fact_4741_UN__finite__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),C5: set(A)] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),C5)
     => aa(set(A),$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),A3),top_top(set(nat))))),C5) ) ).

% UN_finite_subset
tff(fact_4742_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_4743_UN__finite2__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_4744_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B3: 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),A3),C5))),B3) = $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))),B3),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_nd(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C5))) ).

% INT_extend_simps(3)
tff(fact_4745_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_4746_UN__finite2__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A3),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B3),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_4747_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_ni(nat,fun(nat,nat),Nb)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_4748_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_4749_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_4750_cclfp__def,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,A)] : order_532582986084564980_cclfp(A,F2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_nj(fun(A,A),fun(nat,A),F2)),top_top(set(nat)))) ) ).

% cclfp_def
tff(fact_4751_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_4752_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)),insert2(real),zero_zero(real)),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_4753_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,top_top(fun(A,$o))) ).

% top_set_def
tff(fact_4754_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_4755_infinite__UNIV__int,axiom,
    ~ aa(set(int),$o,finite_finite2(int),top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_4756_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_4757_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_4758_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_4759_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_4760_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_nk($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_4761_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_nl(nat,fun(real,real),Nb),Xb)) ).

% root_def
tff(fact_4762_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F2,R) = collect(product_prod(A,A),product_case_prod(A,A,$o,aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_nm(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F2),R))) ).

% mlex_eq
tff(fact_4763_top1I,axiom,
    ! [A: $tType,Xb: A] : aa(A,$o,top_top(fun(A,$o)),Xb) ).

% top1I
tff(fact_4764_mlex__leq,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),mlex_prod(A,F2,R)) ) ) ).

% mlex_leq
tff(fact_4765_mlex__iff,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),mlex_prod(A,F2,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,Xb) = aa(A,nat,F2,Y) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R) ) ) ) ).

% mlex_iff
tff(fact_4766_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),mlex_prod(A,F2,R)) ) ).

% mlex_less
tff(fact_4767_in__measure,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),measure(A,F2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y)) ) ).

% in_measure
tff(fact_4768_in__finite__psubset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A3),B3),finite_psubset(A))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
        & aa(set(A),$o,finite_finite2(A),B3) ) ) ).

% in_finite_psubset
tff(fact_4769_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( 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_4770_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_4771_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Xb: A,S: set(A),Ta: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,Xb,S))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S)
           => has_field_derivative(A,F2,Y,topolo174197925503356063within(A,Xb,Ta)) ) ) ) ).

% has_field_derivative_subset
tff(fact_4772_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F9: A,Xb: A,S: set(A),Ta: set(A)] :
          ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,Xb,S))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S)
           => has_field_derivative(A,F2,F9,topolo174197925503356063within(A,Xb,Ta)) ) ) ) ).

% DERIV_subset
tff(fact_4773_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F9: A,F4: filter(A),G: fun(A,A),G4: A] :
          ( has_field_derivative(A,F2,F9,F4)
         => ( has_field_derivative(A,G,G4,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nn(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F9),G4),F4) ) ) ) ).

% field_differentiable_add
tff(fact_4774_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,Xb: A,S: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_no(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult
tff(fact_4775_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_no(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xb)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult'
tff(fact_4776_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_nn(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D5),E4),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_add
tff(fact_4777_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_np(A,fun(A,A),K),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_4778_at__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [Xb: A,S2: set(A)] : topolo174197925503356063within(A,Xb,S2) = bot_bot(filter(A)) ) ).

% at_discrete
tff(fact_4779_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [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),D4)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3))),aa(real,real,F2,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_4780_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [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),D4)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_4781_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [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),D4)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))),aa(real,real,F2,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_4782_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [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),D4)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_4783_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,Xb: real,Y: real,F2: 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,Y,set_or5935395276787703475ssThan(real,A2,B2))
         => ( ! [X3: real] :
                ( member(real,X3,set_or5935395276787703475ssThan(real,A2,B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) )
           => ( aa(real,real,F2,Xb) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_4784_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Xb: A,Z: A] :
          ( has_field_derivative(A,F2,Y,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_nq(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_4785_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,F2,Y,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_nr(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_at_within_shift
tff(fact_4786_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: 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)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_4787_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: 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)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_4788_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [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),D4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_4789_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [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),D4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))),aa(real,real,F2,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_4790_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [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),D4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3))),aa(real,real,F2,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_4791_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [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),D4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_4792_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ns(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xb)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,Xb)),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_4793_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_nt(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xb))),D5)),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xb)))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_4794_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_4795_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),Ta: set(A),Xb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),Ta)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),topolo174197925503356063within(A,Xb,S)),topolo174197925503356063within(A,Xb,Ta)) ) ) ).

% at_le
tff(fact_4796_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_4797_MVT2,axiom,
    ! [A2: real,B2: real,F2: 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,F2,aa(real,real,F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),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,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(real,real,F9,Z2)) ) ) ) ) ).

% MVT2
tff(fact_4798_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(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,real,F2,Xb) = aa(real,real,F2,Y3) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_4799_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_4800_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_nu(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xaa),K))),topolo174197925503356063within(A,Xaa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_4801_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_nv(fun(A,A),fun(nat,fun(A,A)),F2),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,F2,Xb)),Nb))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power_Suc
tff(fact_4802_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_4803_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_nw(fun(A,A),fun(nat,fun(A,A)),F2),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,F2,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_4804_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(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),aa(real,real,F2,Y3)),aa(real,real,F2,Xb)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_4805_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(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),aa(real,real,F2,Xb)),aa(real,real,F2,Y3)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_4806_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_4807_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_nx(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_4808_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_4809_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_4810_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_4811_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ns(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),E2),aa(A,A,F2,Xb))),aa(nat,A,power_power(A,aa(A,A,G,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_4812_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_nt(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,aa(A,A,F2,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_4813_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),F2: fun(A,A),F9: A,Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(A,fun(nat,A)),C2),Z2),aa(A,A,F2,Z2)) )
         => ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_ny(fun(nat,A),fun(A,fun(nat,A)),C2),Z),F9) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_4814_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_nz(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_4815_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ny(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_oa(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
             => ( aa(real,$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_ob(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ny(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_4816_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$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_ob(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ny(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_4817_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
           => has_field_derivative(A,aTP_Lamp_ob(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ny(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_4818_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_4819_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),Mb: real,Xb: real,R2: real] :
      ( has_field_derivative(real,G,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,G,Xb))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_oc(fun(real,real),fun(real,fun(real,real)),G),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G,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_4820_DERIV__powr,axiom,
    ! [G: fun(real,real),Mb: real,Xb: real,F2: fun(real,real),R2: real] :
      ( has_field_derivative(real,G,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,G,Xb))
       => ( has_field_derivative(real,F2,R2,topolo174197925503356063within(real,Xb,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_od(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,Xb),aa(real,real,F2,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,G,Xb)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),Mb),aa(real,real,F2,Xb)),aa(real,real,G,Xb)))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_4821_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_4822_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: 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,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_4823_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_4824_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_4825_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Xb: A,Db: A,S: set(A)] :
          ( ( cosh(A,aa(A,A,G,Xb)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aTP_Lamp_oe(fun(A,A),fun(A,A),G),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,G,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_4826_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_4827_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: 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,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_4828_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun(nat,real),X0: real] :
      ( ! [X3: real] :
          ( member(real,X3,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_of(fun(nat,real),fun(real,fun(nat,real)),F2),X3)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => has_field_derivative(real,aTP_Lamp_oh(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_of(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_4829_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_4830_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_4831_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xb: real,Nb: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M2: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ? [T6: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xb))
          & ( aa(real,real,F2,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_oi(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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_4832_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xb: real,Nb: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M2: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F2,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_oi(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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_4833_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ 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_4834_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M2: nat,T6: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),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),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
           => ? [T6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T6)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),H)
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_oj(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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_4835_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: 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)) = F2 )
       => ( ! [M2: nat,T6: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),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),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ? [T6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T6)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),H)
              & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_oj(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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_4836_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),zero_zero(real))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M2: nat,T6: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H),T6)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
           => ? [T6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T6)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),zero_zero(real))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_oj(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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_4837_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( Xb != zero_zero(real) )
         => ( ! [M2: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
           => ? [T6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T6))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xb))
                & ( aa(real,real,F2,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_oi(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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_4838_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M2: nat,T6: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xb)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
       => ? [T6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T6)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F2,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_oi(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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_4839_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: 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)) = F2 )
       => ( ! [M2: nat,T6: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T6)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,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_eq(real),C2),B2)
             => ? [T6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),T6)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),C2)
                  & ( aa(real,real,F2,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_ok(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_4840_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: 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)) = F2 )
       => ( ! [M2: nat,T6: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T6)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),B2)
             => ? [T6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T6)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),B2)
                  & ( aa(real,real,F2,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_ok(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_4841_Taylor,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: 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)) = F2 )
       => ( ! [M2: nat,T6: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T6)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),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 != C2 )
                   => ? [T6: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),C2),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T6)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),C2) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T6)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T6),Xb) ) )
                        & ( aa(real,real,F2,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_ol(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),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),T6),semiring_char_0_fact(real,Nb))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),C2)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_4842_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),$o,aTP_Lamp_om(set(A),fun(set(A),$o)))) ).

% finite_psubset_def
tff(fact_4843_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B3: real] :
      ( ! [M2: nat,T6: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),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),Diff,M2),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6),topolo174197925503356063within(real,T6,top_top(set(real)))) )
     => ( ( Nb = aa(nat,nat,suc,K) )
       => ! [M: nat,T7: 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)),T7)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),H) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_oo(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Nb),Diff),B3),M),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T7)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_op(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M),T7)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M))))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),divide_divide(real,aa(nat,real,power_power(real,T7),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M))))))),topolo174197925503356063within(real,T7,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_4844_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_oq(real,real),suminf(real,aTP_Lamp_or(real,fun(nat,real),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_4845_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) )
       => ( ( 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)))))) ) ) )
         => ( ( 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))))))) ) ) )
           => ( ( ~ 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_4846_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_4847_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G4: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_os(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ot(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G4),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_4848_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G4: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_ou(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ov(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G4),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_4849_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F9: fun(A,real),Xb: A,S: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,real,F2,F9,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ow(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ox(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F9),Xb),G),G4),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_4850_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_oy(B,fun(A,B),C2),aTP_Lamp_oz(A,B),F4) ) ).

% has_derivative_const
tff(fact_4851_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),F4: filter(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,F4)
         => ( has_derivative(A,B,G,G4,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pa(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_pa(fun(A,B),fun(fun(A,B),fun(A,B)),F9),G4),F4) ) ) ) ).

% has_derivative_add
tff(fact_4852_has__derivative__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A,S: set(A),Ta: set(A)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,S))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S)
           => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,Ta)) ) ) ) ).

% has_derivative_subset
tff(fact_4853_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_oz(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_4854_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A,S: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pb(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pc(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F9),Xb),G),G4),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_mult
tff(fact_4855_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Ta: set(A),G: fun(A,B),G4: fun(A,fun(A,B)),F2: fun(C,A),S: set(C),Xb: C,F9: fun(C,A)] :
          ( ! [X3: A] :
              ( member(A,X3,Ta)
             => has_derivative(A,B,G,aa(A,fun(A,B),G4,X3),topolo174197925503356063within(A,X3,Ta)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),S)),Ta)
           => ( member(C,Xb,S)
             => ( has_derivative(C,A,F2,F9,topolo174197925503356063within(C,Xb,S))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pd(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_pe(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G4),F2),Xb),F9),topolo174197925503356063within(C,Xb,S)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_4856_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A,S2: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,S2))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xb,S2))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pg(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F9),Xb),G),G4),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_4857_arccos__less__arccos,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),aa(real,real,arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_4858_arccos__less__mono,axiom,
    ! [Xb: real,Y: 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),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Xb)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_4859_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),Xb: B,F9: fun(B,A),S2: set(B)] :
          ( ( aa(B,A,F2,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F9,topolo174197925503356063within(B,Xb,S2))
           => has_derivative(B,A,aTP_Lamp_ph(fun(B,A),fun(B,A),F2),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_pi(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),Xb),F9),topolo174197925503356063within(B,Xb,S2)) ) ) ) ).

% has_derivative_inverse
tff(fact_4860_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_pj(A,fun(A,A),Xb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_inverse'
tff(fact_4861_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_4862_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G4: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_pk(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pl(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G4),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_ln
tff(fact_4863_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_4864_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G4: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_pm(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_pn(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G4),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_4865_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A,S2: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,S2))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xb,S2))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_po(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_pp(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F9),Xb),G),G4),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% has_derivative_divide
tff(fact_4866_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F9: fun(A,fun(B,C)),Xb: B,S2: set(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I2),aa(A,fun(B,C),F9,I2),topolo174197925503356063within(B,Xb,S2)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_pr(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_pt(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),F2),F9),Xb),topolo174197925503356063within(B,Xb,S2)) ) ) ).

% has_derivative_prod
tff(fact_4867_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),Xb: A,X5: set(A),F2: fun(A,real),F9: fun(A,real)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xb,X5))
         => ( has_derivative(A,real,F2,F9,topolo174197925503356063within(A,Xb,X5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
             => ( member(A,Xb,X5)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pu(fun(A,real),fun(fun(A,real),fun(A,real)),G),F2),aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_pv(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G4),Xb),F2),F9),topolo174197925503356063within(A,Xb,X5)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_4868_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_oa(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$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_px(fun(nat,A),fun(A,fun(A,A)),C2),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_4869_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_py(nat,set(nat))),M4))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_4870_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),C2: fun(nat,B),K5: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_pz(fun(nat,B),fun(B,fun(nat,B)),C2),K5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,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_qb(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_4871_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)),insert2(A),Xb),bot_bot(set(A)))) = Xb ) ).

% Max_singleton
tff(fact_4872_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aTP_Lamp_aq(nat,fun(nat,$o),Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_4873_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xb)
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xb) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_4874_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xb)
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Xb) ) ) ) ) ) ).

% Max_less_iff
tff(fact_4875_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qc(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_4876_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_qd(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_4877_power__tendsto__0__iff,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$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_qe(nat,fun(fun(A,real),fun(A,real)),Nb),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_4878_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A3: set(A),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_qf(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_4879_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ).

% Max_insert
tff(fact_4880_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qg(A,fun(fun(A,B),fun(A,B)),Xb),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xb)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_4881_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,A2: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_qh(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_4882_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_qi(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ).

% tendsto_arcosh
tff(fact_4883_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_qj(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F4) ) ) ) ).

% tendsto_cot
tff(fact_4884_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( cosh(B,A2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_qk(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A2)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_4885_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_ql(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F4) ) ) ) ).

% tendsto_tan
tff(fact_4886_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_qm(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_qn(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_4887_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_qo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_4888_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_qp(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A2)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_4889_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_qq(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_4890_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_qq(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_4891_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_qq(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_4892_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_4893_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qs(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_4894_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,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)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_4895_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,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)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_4896_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qv(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_4897_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qv(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_4898_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A2: B,F4: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qw(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform
tff(fact_4899_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform2
tff(fact_4900_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_qv(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_4901_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F4: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_4902_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,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)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_4903_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_qz(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,divide_divide(B,A2,B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_4904_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,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)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),F4) ) ) ) ).

% tendsto_add
tff(fact_4905_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D2: B,F4: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ra(B,fun(fun(A,B),fun(A,B)),C2),F2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),D2)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D2),F4) ) ) ).

% tendsto_add_const_iff
tff(fact_4906_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_rb(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sgn_sgn(B,L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_4907_tendsto__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F4: filter(A),F2: fun(A,B),A2: B,B2: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
           => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),F4)
             => ( A2 = B2 ) ) ) ) ) ).

% tendsto_unique
tff(fact_4908_tendsto__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),A2: B] : filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),bot_bot(filter(A))) ) ).

% tendsto_bot
tff(fact_4909_nhds__neq__bot,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo7230453075368039082e_nhds(A,A2) != bot_bot(filter(A)) ) ).

% nhds_neq_bot
tff(fact_4910_continuous__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B)] : topolo3448309680560233919inuous(A,B,bot_bot(filter(A)),F2) ) ).

% continuous_bot
tff(fact_4911_continuous__trivial__limit,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Net: filter(A),F2: fun(A,B)] :
          ( ( Net = bot_bot(filter(A)) )
         => topolo3448309680560233919inuous(A,B,Net,F2) ) ) ).

% continuous_trivial_limit
tff(fact_4912_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_rc(B,fun(A,B),A2),topolo7230453075368039082e_nhds(B,B2),F4)
          <=> ( A2 = B2 ) ) ) ) ).

% tendsto_const_iff
tff(fact_4913_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$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))),D3) )
                     => ( aa(A,B,F2,X3) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rd(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_4914_tendsto__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),L: filter(B),Xb: A,S2: set(A),T3: set(A)] :
          ( filterlim(A,B,F2,L,topolo174197925503356063within(A,Xb,S2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S2)
           => filterlim(A,B,F2,L,topolo174197925503356063within(A,Xb,T3)) ) ) ) ).

% tendsto_within_subset
tff(fact_4915_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_re(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_4916_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_re(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_4917_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_re(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_4918_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_rf(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_4919_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_rg(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_4920_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( aa(real,$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_rh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_4921_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ).

% Max_ge
tff(fact_4922_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [Y3: A] :
                ( member(A,Y3,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
           => ( member(A,Xb,A3)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = Xb ) ) ) ) ) ).

% Max_eqI
tff(fact_4923_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ! [X3: A] :
                  ( member(A,X3,A3)
                 => ? [Xa: A] :
                      ( member(A,Xa,B3)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,B3)
                   => ? [Xa: A] :
                        ( member(A,Xa,A3)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,lattic643756798349783984er_Max(A),B3) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_4924_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ).

% Max.coboundedI
tff(fact_4925_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A3),A3) ) ) ) ).

% Max_in
tff(fact_4926_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$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_ri(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_4927_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ).

% Max.in_idem
tff(fact_4928_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_rj(B,fun(fun(B,C),fun(B,C)),A2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_4929_IVT,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),A2: B,Y: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,A2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,B2))
           => ( aa(B,$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,$o),ord_less_eq(B),X3),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X3,top_top(set(B))),F2) )
               => ? [X3: B] :
                    ( aa(B,$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,F2,X3) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_4930_IVT2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),B2: B,Y: A,A2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,B2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,A2))
           => ( aa(B,$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,$o),ord_less_eq(B),X3),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X3,top_top(set(B))),F2) )
               => ? [X3: B] :
                    ( aa(B,$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,F2,X3) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_4931_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$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,F2,X)),L5))),R2) ) ) ) ) ) ).

% LIM_D
tff(fact_4932_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),S7) )
                     => aa(real,$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,F2,X3)),L5))),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_4933_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A2))),S6) )
                     => aa(real,$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,F2,X2)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_4934_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$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,F2,X3) = aa(A,B,G,X3) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_4935_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D5: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rk(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rl(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_4936_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( 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 != C2 )
                  & 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),C2),X))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_4937_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X: real] :
                ( ( ( X != C2 )
                  & 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),C2),X))),R3) )
               => ( aa(real,real,F2,X) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_4938_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( 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 != C2 )
                  & 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),C2),X))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F2,X)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_4939_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$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))),D3) )
                     => ( aa(A,B,F2,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rd(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_4940_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_4941_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_qo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_4942_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_rn(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_4943_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aTP_Lamp_ro(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_4944_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A3)
                 => 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),A3)),Xb) ) ) ) ) ).

% Max.boundedI
tff(fact_4945_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xb)
             => ! [A9: A] :
                  ( member(A,A9,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),Xb) ) ) ) ) ) ).

% Max.boundedE
tff(fact_4946_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798349783984er_Max(A),A3) )
            <=> ( member(A,Mb,A3)
                & ! [X2: A] :
                    ( member(A,X2,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Mb) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4947_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3))
            <=> ? [X2: A] :
                  ( member(A,X2,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X2) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_4948_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = Mb )
            <=> ( member(A,Mb,A3)
                & ! [X2: A] :
                    ( member(A,X2,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Mb) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4949_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3))
            <=> ? [X2: A] :
                  ( member(A,X2,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X2) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_4950_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [B4: A] :
                ( member(A,B4,A3)
               => 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)),insert2(A),A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_4951_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_4952_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).

% Max_Sup
tff(fact_4953_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rp(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_4954_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rp(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_4955_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_rq(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_4956_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( ! [H4: A] :
                ( ( H4 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H4)),K)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,H4))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H4))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_4957_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$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,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M7: A] :
              ! [X: real] :
                ( ( aa(real,$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,F2,X)),M7) ) ) ) ) ).

% isCont_bounded
tff(fact_4958_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$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,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$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,$o),ord_less_eq(A),aa(real,A,F2,X)),M7) )
                & ? [X3: real] :
                    ( aa(real,$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,F2,X3) = M7 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_4959_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$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,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$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,$o),ord_less_eq(A),M7),aa(real,A,F2,X)) )
                & ? [X3: real] :
                    ( aa(real,$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,F2,X3) = M7 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_4960_isCont__inverse__function2,axiom,
    ! [A2: real,Xb: real,B2: real,G: fun(real,real),F2: 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,G,aa(real,real,F2,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))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xb),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_4961_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_derivative(A,A,F2,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_rp(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_4962_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( ( aa(A,B,G,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_4963_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_ro(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_4964_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_rr(fun(A,B),fun(A,fun(A,B)),F2),A2),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_4965_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),F2)
         => ( ( cos(A,aa(A,A,F2,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_ql(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_4966_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_4967_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B3)) ) ) ) ) ).

% Max.subset_imp
tff(fact_4968_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),F2)
         => ( ( sin(A,aa(A,A,F2,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_qj(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_4969_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Xb: A,A3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A3),F2)
         => ( ( cosh(B,aa(A,B,F2,Xb)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A3),aTP_Lamp_rs(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_4970_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( aa(set(A),$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_4971_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B3)),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ) ).

% Max.subset
tff(fact_4972_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_4973_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A3),A3) ) ) ) ) ).

% Max.closed
tff(fact_4974_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_4975_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_4976_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      divide_divide(nat,Mb,Nb) = $ite(Nb = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_rt(nat,fun(nat,fun(nat,$o)),Mb),Nb)))) ).

% divide_nat_def
tff(fact_4977_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S2: set(A),F2: fun(A,B),K: B] :
          ( aa(set(A),$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_ru(fun(A,B),fun(B,fun(A,B)),F2),K)),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,F2),S2))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_4978_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$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,$o),ord_less_eq(real),X3),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$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,$o),ord_less_eq(A),aa(real,A,F2,X)),M7) )
                & ! [N7: A] :
                    ( aa(A,$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,F2,X3)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_4979_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_4980_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_4981_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_4982_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$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_hl(fun(nat,A),fun(A,fun(nat,A)),A2),X3),aa(A,A,F2,X3)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_4983_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$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_hl(fun(nat,A),fun(A,fun(nat,A)),A2),X3),aa(A,A,F2,X3)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_4984_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( summable(real,F2)
           => ( ! [H4: A,N: nat] :
                  ( ( H4 != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H4)),K)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H4),N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N)),real_V7770717601297561774m_norm(A,H4))) ) )
             => filterlim(A,B,aTP_Lamp_rv(fun(A,fun(nat,B)),fun(A,B),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_4985_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( cos(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_ql(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_4986_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_4987_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A2))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),aa(fun(A,real),fun(A,real),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_4988_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( ( sin(A,aa(A,A,F2,A2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_qj(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_4989_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_4990_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_4991_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D5: real,G: fun(real,real),Xb: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D5,topolo174197925503356063within(real,aa(real,real,G,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)
           => ( ! [Y3: real] :
                  ( 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)
                   => ( aa(real,real,F2,aa(real,real,G,Y3)) = Y3 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D5),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_4992_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_4993_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_4994_LIM__less__bound,axiom,
    ! [B2: real,Xb: real,F2: 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,F2,X3)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_4995_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A2))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_4996_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_4997_sum__le__card__Max,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image2(A,nat,F2),A3)))) ) ).

% sum_le_card_Max
tff(fact_4998_isCont__inverse__function,axiom,
    ! [D2: real,Xb: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
     => ( ! [Z2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z2),Xb))),D2)
           => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z2),Xb))),D2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xb),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_4999_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real),G4: fun(real,real),F9: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),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))),F2) ) )
       => ( ! [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))),G) ) )
         => ( ! [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,G,aa(real,real,G4,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B2)
                   => has_field_derivative(real,F2,aa(real,real,F9,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C4),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,G4,C4)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F9,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_5000_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$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_ob(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_5001_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_rx(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_rx(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_rx(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_5002_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_rx(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_rx(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_rx(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_5003_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_ry(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_rx(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_5004_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_5005_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),collect(int,aTP_Lamp_aj(int,fun(int,$o),Nb))) = aa(int,int,abs_abs(int),Nb) ) ) ).

% Max_divisors_self_int
tff(fact_5006_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rz(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_5007_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sa(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_5008_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sb(A,fun(fun(nat,A),fun(nat,A)),C2),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_5009_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_5010_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [U3: fun(nat,A)] :
              ( ! [N4: nat] : aa(A,$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_5011_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ? [U3: fun(nat,A)] :
              ( ! [N4: nat] : aa(A,$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_5012_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),K: nat,A2: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_5013_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),A2: A,K: nat] :
          ( filterlim(nat,A,F2,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)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_5014_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N3: nat,X5: fun(nat,A),Y5: fun(nat,A),Xb: A,Y: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,Y5,N)) )
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( filterlim(nat,A,Y5,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ) ).

% lim_mono
tff(fact_5015_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,Y5: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( filterlim(nat,A,Y5,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N7: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,Y5,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ) ).

% LIMSEQ_le
tff(fact_5016_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M4: nat,C5: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),C5) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C5) ) ) ) ).

% Lim_bounded
tff(fact_5017_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N3: nat,C5: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),aa(nat,A,F2,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),L) ) ) ) ).

% Lim_bounded2
tff(fact_5018_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,A2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N7: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,X5,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb) ) ) ) ).

% LIMSEQ_le_const
tff(fact_5019_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,A2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N7: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_5020_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N: nat] : member(A,aa(nat,A,B2,N),S)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,complete_Sup_Sup(A),S)) ) ) ) ).

% Sup_lim
tff(fact_5021_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N: nat] : member(A,aa(nat,A,B2,N),S)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S)),A2) ) ) ) ).

% Inf_lim
tff(fact_5022_Inf__as__limit,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ? [U3: fun(nat,A)] :
              ( ! [N4: nat] : member(A,aa(nat,A,U3,N4),A3)
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Inf_Inf(A),A3)),at_top(nat)) ) ) ) ).

% Inf_as_limit
tff(fact_5023_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_5024_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => filterlim(nat,nat,aTP_Lamp_sd(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_5025_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_5026_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,$o),ord_less_eq(A),aa(nat,A,A2,N4)),Xb)
                & ! [M: nat,N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,M)),aa(nat,A,A2,N4)) ) )
              | ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,A2,N4))
                & ! [M: nat,N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N4)),aa(nat,A,A2,M)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_5027_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_5028_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_5029_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_5030_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R3),aa(nat,real,X5,N)) )
     => 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_5031_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
     => filterlim(nat,real,aTP_Lamp_si(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_5032_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),L)
       => ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [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,F2,N4)),E)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_5033_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_5034_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_5035_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_sk(fun(nat,A),fun(nat,A),F2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_5036_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_sl(fun(nat,A),fun(nat,A),F2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_5037_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_5038_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real))
     => filterlim(nat,real,power_power(real,aa(real,real,abs_abs(real),C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_5039_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real))
     => filterlim(nat,real,power_power(real,C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_5040_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_5041_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( sums(A,F2,S)
        <=> filterlim(nat,A,aTP_Lamp_so(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_5042_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),Xb: real] :
          ( filterlim(nat,real,aTP_Lamp_sp(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Xb),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_5043_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,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N4)
               => aa(real,$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_5044_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N)
                 => aa(real,$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))),R3) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_5045_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,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N2)
                 => aa(real,$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))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_5046_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_5047_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F4: filter(A),Xb: B] :
          ( filterlim(A,nat,F2,at_top(nat),F4)
         => ( aa(real,$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)),F2),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_5048_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_5049_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N: nat] : aa(nat,A,S,N) != 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))),F2),Z),S),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_5050_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_5051_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_5052_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_hs(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_5053_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( 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_rx(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_rx(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_ry(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_5054_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_ry(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_rx(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_5055_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))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aTP_Lamp_rx(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_rx(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_5056_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,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_sv(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_5057_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D5: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,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_sw(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D5),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_5058_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,F2,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_sv(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_within
tff(fact_5059_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_oz(A,B)) ) ).

% bounded_linear_zero
tff(fact_5060_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pa(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_5061_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sx(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_5062_has__derivative__within__singleton__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,G,topolo174197925503356063within(A,Xb,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))
        <=> real_V3181309239436604168linear(A,B,G) ) ) ).

% has_derivative_within_singleton_iff
tff(fact_5063_filterlim__pow__at__top,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qe(nat,fun(fun(A,real),fun(A,real)),Nb),F2),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_5064_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_5065_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K5: real] :
          ( ! [X3: A,Y3: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3))
         => ( ! [R3: real,X3: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R3),X3)) = aa(B,B,real_V8093663219630862766scaleR(B,R3),aa(A,B,F2,X3))
           => ( ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K5))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_5066_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sy(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_5067_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_5068_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F2: fun(A,real),F4: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),S),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ta(real,fun(fun(A,real),fun(A,real)),S),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_neg_powr
tff(fact_5069_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),X3)
         => ? [Y2: real] :
              ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),zero_zero(real)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_5070_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,F2,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_tb(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_5071_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F9: fun(A,B),Xb: A,F2: fun(A,B),S: set(A)] :
          ( real_V3181309239436604168linear(A,B,F9)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_tc(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F9),Xb),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivativeI
tff(fact_5072_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & ? [E3: fun(A,B)] :
                ( ! [H5: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),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,F2,Xb)),aa(A,B,F9,H5))),aa(A,B,E3,H5))
                & filterlim(A,real,aTP_Lamp_td(fun(A,B),fun(A,real),E3),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_5073_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F9,F4)
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F9),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_5074_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xb: A,S2: set(A),F2: fun(A,B),F9: fun(A,B)] :
          ( member(A,Xb,S2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,S2))
            <=> ( real_V3181309239436604168linear(A,B,F9)
                & ? [E3: fun(A,B)] :
                    ( ! [H5: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5),S2)
                       => ( aa(A,B,F2,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,F2,Xb)),aa(A,B,F9,H5))),aa(A,B,E3,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_td(fun(A,B),fun(A,real),E3),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_5075_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F9: fun(A,B),S: set(A),Xb: A,F2: fun(A,B),H6: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ( real_V3181309239436604168linear(A,B,F9)
           => ( ! [Y3: A] :
                  ( member(A,Y3,S)
                 => ( ( Y3 != Xb )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y3,Xb)),E2)
                     => aa(real,$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,F2,Y3)),aa(A,B,F2,Xb))),aa(A,B,F9,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xb)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xb)))),aa(A,real,H6,Y3)) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S))
               => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_5076_open__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1002775350975398744n_open(A,bot_bot(set(A))) ) ).

% open_empty
tff(fact_5077_dist__add__cancel2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [B2: A,A2: A,C2: 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),C2),A2)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel2
tff(fact_5078_dist__add__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: 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),C2)) = real_V557655796197034286t_dist(A,B2,C2) ) ).

% dist_add_cancel
tff(fact_5079_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_5080_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Y))
        <=> ( Xb != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_5081_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_5082_open__INT,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A3: set(A),B3: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => topolo1002775350975398744n_open(B,aa(A,set(B),B3,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),B3),A3))) ) ) ) ).

% open_INT
tff(fact_5083_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,D2: real] : topolo1002775350975398744n_open(A,collect(A,aa(real,fun(A,$o),aTP_Lamp_tg(A,fun(real,fun(A,$o)),Xb),D2))) ) ).

% open_ball
tff(fact_5084_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X2: A] :
              ( member(A,X2,S2)
             => ? [E3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
                  & ! [Y4: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y4,X2)),E3)
                     => member(A,Y4,S2) ) ) ) ) ) ).

% open_dist
tff(fact_5085_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,Xb: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,Xb)),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),E2) ) ) ).

% dist_commute_lessI
tff(fact_5086_openI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( ! [X3: A] :
              ( member(A,X3,S2)
             => ? [T8: set(A)] :
                  ( topolo1002775350975398744n_open(A,T8)
                  & member(A,X3,T8)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T8),S2) ) )
         => topolo1002775350975398744n_open(A,S2) ) ) ).

% openI
tff(fact_5087_open__subopen,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X2: A] :
              ( member(A,X2,S2)
             => ? [T9: set(A)] :
                  ( topolo1002775350975398744n_open(A,T9)
                  & member(A,X2,T9)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),S2) ) ) ) ) ).

% open_subopen
tff(fact_5088_first__countable__basis,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Xb: A] :
        ? [A6: fun(nat,set(A))] :
          ( ! [I3: nat] :
              ( member(A,Xb,aa(nat,set(A),A6,I3))
              & topolo1002775350975398744n_open(A,aa(nat,set(A),A6,I3)) )
          & ! [S8: set(A)] :
              ( ( topolo1002775350975398744n_open(A,S8)
                & member(A,Xb,S8) )
             => ? [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),A6,I2)),S8) ) ) ) ).

% first_countable_basis
tff(fact_5089_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_5090_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Y)) ) ) ).

% dist_pos_lt
tff(fact_5091_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_5092_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xb) )
           => ~ member(A,aa(set(A),A,complete_Sup_Sup(A),A3),A3) ) ) ) ).

% Sup_notin_open
tff(fact_5093_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X3) )
           => ~ member(A,aa(set(A),A,complete_Inf_Inf(A),A3),A3) ) ) ) ).

% Inf_notin_open
tff(fact_5094_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)),insert2(A),Xb),bot_bot(set(A)))) ) ).

% not_open_singleton
tff(fact_5095_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y: A,E1: real,X22: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,Y)),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_5096_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Z: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xb,Z)),real_V557655796197034286t_dist(A,Y,Z))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),E2) ) ) ).

% dist_triangle_lt
tff(fact_5097_hausdorff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ? [U4: set(A),V5: set(A)] :
              ( topolo1002775350975398744n_open(A,U4)
              & topolo1002775350975398744n_open(A,V5)
              & member(A,Xb,U4)
              & member(A,Y,V5)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),V5) = bot_bot(set(A)) ) ) ) ) ).

% hausdorff
tff(fact_5098_separation__t2,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
        <=> ? [U5: set(A),V6: set(A)] :
              ( topolo1002775350975398744n_open(A,U5)
              & topolo1002775350975398744n_open(A,V6)
              & member(A,Xb,U5)
              & member(A,Y,V6)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V6) = bot_bot(set(A)) ) ) ) ) ).

% separation_t2
tff(fact_5099_at__within__open__subset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S2: set(A),T3: set(A)] :
          ( member(A,A2,S2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
             => ( topolo174197925503356063within(A,A2,T3) = topolo174197925503356063within(A,A2,top_top(set(A))) ) ) ) ) ) ).

% at_within_open_subset
tff(fact_5100_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),Xb: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( member(A,Xb,S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
             => ? [B4: A] :
                  ( aa(A,$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_5101_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_th(A,A)) = Xb ) ) ) ).

% Lim_ident_at
tff(fact_5102_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F9: A,A2: A,S2: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,A2,S2))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,A2,S2)
             => ( ! [X3: A] :
                    ( member(A,X3,S2)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D2)
                     => ( aa(A,A,F2,X3) = aa(A,A,G,X3) ) ) )
               => has_field_derivative(A,G,F9,topolo174197925503356063within(A,A2,S2)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_5103_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xb: A,S: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xb,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,Xb,S)
             => ( ! [X7: A] :
                    ( member(A,X7,S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X7,Xb)),D2)
                     => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) )
               => has_derivative(A,B,G,F9,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_5104_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M8: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M3),aa(nat,A,X5,N2))),E3) ) ) ) ) ) ).

% Cauchy_def
tff(fact_5105_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N6: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S,N2),aa(nat,A,S,N6))),E3) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_5106_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M7: nat] :
              ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M)
               => ! [N4: nat] :
                    ( aa(nat,$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,M),aa(nat,A,X5,N4))),E2) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_5107_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M9: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M2),aa(nat,A,X5,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% metric_CauchyI
tff(fact_5108_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S9: set(A)] :
              ( topolo1002775350975398744n_open(A,S9)
             => ( member(A,F0,S9)
               => ? [N6: nat] :
                  ! [N2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N2)
                   => member(A,aa(nat,A,F2,N2),S9) ) ) ) ) ) ).

% lim_explicit
tff(fact_5109_tendsto__Lim,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Net: filter(A),F2: fun(A,B),L: B] :
          ( ( Net != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net)
           => ( topolo3827282254853284352ce_Lim(A,B,Net,F2) = L ) ) ) ) ).

% tendsto_Lim
tff(fact_5110_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_rm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_5111_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_rn(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_5112_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_ro(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_5113_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,Xb: A,S2: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xb,S2))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( ! [X7: A] :
                  ( member(A,X7,S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X7,Xb))
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X7,Xb)),D2)
                     => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% Lim_transform_within
tff(fact_5114_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y: A,E2: real,X22: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,Y)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),E2) ) ) ) ).

% dist_triangle_half_l
tff(fact_5115_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X15: A,E2: real,X22: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X15)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X22)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),E2) ) ) ) ).

% dist_triangle_half_r
tff(fact_5116_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,X22: A,E2: real,X32: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit1(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X22,X32)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit1(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit1(one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X42)),E2) ) ) ) ) ).

% dist_triangle_third
tff(fact_5117_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A,S2: set(A),T3: 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)),T3),S2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(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)),insert2(A),Xb),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,Xb,T3) = topolo174197925503356063within(A,Xb,U2) ) ) ) ) ) ).

% at_within_nhd
tff(fact_5118_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G3: filter(B),Xb: A,S2: set(A),F4: filter(B),D2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G3,topolo174197925503356063within(A,Xb,S2))
         => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G3),F4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
             => ( ! [X7: A] :
                    ( member(A,X7,S2)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X7,Xb))
                     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X7,Xb)),D2)
                       => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) ) )
               => filterlim(A,B,F2,F4,topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_5119_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M9: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M2)
                 => ! [N: nat] :
                      ( aa(nat,$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,X5,M2),aa(nat,A,X5,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI'
tff(fact_5120_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M8: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M3)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M3),aa(nat,A,F2,N2))),E3) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_5121_at__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] :
          ( ( topolo174197925503356063within(A,A2,top_top(set(A))) = bot_bot(filter(A)) )
        <=> topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ) ).

% at_eq_bot_iff
tff(fact_5122_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),S6) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_5123_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$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,F2,X),L5)),R2) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_5124_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X3),L5)),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_5125_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$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,F2,X3) = aa(A,B,G,X3) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_5126_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,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N2),L5)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_5127_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,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N),L5)),R3) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_5128_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,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N4)
               => aa(real,$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_5129_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [J3: nat] :
            ? [M8: nat] :
            ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M3)
             => ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M3),aa(nat,A,X5,N2))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_5130_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D3) )
                     => ( aa(A,B,F2,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ti(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_5131_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_ql(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_5132_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_tj(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_qj(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_5133_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( cosh(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_rs(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_5134_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_tk(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_5135_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D3) )
                     => ( aa(A,B,F2,X3) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ti(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_5136_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S2: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( member(B,A2,S2)
           => ( topolo1002775350975398744n_open(B,S2)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,S2))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_rj(B,fun(fun(B,C),fun(B,C)),A2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_5137_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [K2: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),K2)
                  & 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_tm(real,fun(A,set(A)),E3)),K2))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_5138_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_rw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_5139_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,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),No3)
                  & ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N2),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_5140_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_tn(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_5141_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_to(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_5142_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_tn(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_5143_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_5144_not__tendsto__and__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F4: filter(A),F2: fun(A,B),C2: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
           => ~ filterlim(A,B,F2,at_infinity(B),F4) ) ) ) ).

% not_tendsto_and_filterlim_at_infinity
tff(fact_5145_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tp(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_5146_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tp(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_5147_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_5148_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F4)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_5149_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_5150_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_ts(fun(A,B),fun(nat,fun(A,B)),F2),Nb),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_5151_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_5152_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_5153_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_qp(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_5154_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_5155_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F4: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_ns(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_5156_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_5157_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B2)
         => ? [Y2: real] :
              ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_5158_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,Nb: nat,B3: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_tt(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),Nb),B3),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_5159_GMVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: 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))),F2) )
       => ( ! [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,F2,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))),G) )
           => ( ! [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,G,topolo174197925503356063within(real,X3,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C4),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_5160_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_qp(fun(A,B),fun(A,B),F2),F4) ) ) ) ).

% Bfun_inverse
tff(fact_5161_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_tu(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_5162_eventually__const,axiom,
    ! [A: $tType,F4: filter(A),P: $o] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_nc($o,fun(A,$o),(P)),F4)
      <=> (P) ) ) ).

% eventually_const
tff(fact_5163_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C2: B,Q2: fun(A,B),Ta: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tv(B,fun(fun(A,B),fun(A,B)),C2),Q2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_5164_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q2: fun(A,B),C2: B,Ta: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_tw(fun(A,B),fun(B,fun(A,B)),Q2),C2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_5165_eventually__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] : eventually(A,P,bot_bot(filter(A))) ).

% eventually_bot
tff(fact_5166_eventually__happens,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( eventually(A,P,Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).

% eventually_happens
tff(fact_5167_eventually__happens_H,axiom,
    ! [A: $tType,F4: filter(A),P: fun(A,$o)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,P,F4)
       => ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).

% eventually_happens'
tff(fact_5168_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X3: A] : ~ aa(A,$o,P,X3)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_5169_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F4: filter(A)] :
      ( eventually(A,aTP_Lamp_nc($o,fun(A,$o),(P)),F4)
    <=> ( (P)
        | ( F4 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_5170_trivial__limit__def,axiom,
    ! [A: $tType,F4: filter(A)] :
      ( ( F4 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_am(A,$o),F4) ) ).

% trivial_limit_def
tff(fact_5171_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pa(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_add
tff(fact_5172_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N6: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N6),N2)
             => aa(A,$o,P,N2) ) ) ) ).

% eventually_at_top_linorder
tff(fact_5173_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,$o)] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X3)
             => aa(A,$o,P,X3) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_5174_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N6: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N6),N2)
             => aa(A,$o,P,N2) ) ) ) ).

% eventually_at_top_dense
tff(fact_5175_eventually__sequentially,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N6: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N2)
         => aa(nat,$o,P,N2) ) ) ).

% eventually_sequentially
tff(fact_5176_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,$o)] :
      ( ! [X3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),X3)
         => aa(nat,$o,P,X3) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_5177_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N6: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N2),N6)
             => aa(A,$o,P,N2) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_5178_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N6: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N2),N6)
             => aa(A,$o,P,N2) ) ) ) ).

% eventually_at_bot_dense
tff(fact_5179_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_5180_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_5181_differentiable__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A),Ta: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S)
           => differentiable(A,B,F2,topolo174197925503356063within(A,Xb,Ta)) ) ) ) ).

% differentiable_within_subset
tff(fact_5182_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_5183_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_tu(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_5184_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_tx(A,fun(A,$o)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_5185_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_ty(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_5186_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_tz(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_5187_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_tz(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_5188_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
         => ~ ! [B9: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B9)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(real,fun(A,$o)),F2),B9),F4) ) ) ) ).

% BfunE
tff(fact_5189_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(real,fun(A,$o)),F2),K6),F4) ) ) ) ).

% Bfun_def
tff(fact_5190_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),K: nat] :
          ( bfun(nat,A,X5,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ub(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_5191_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),K: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ub(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat))
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_5192_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [Y4: B,K6: real] :
              ( aa(real,$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,$o)),aTP_Lamp_uc(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),Y4),K6),F4) ) ) ) ).

% Bfun_metric_def
tff(fact_5193_differentiable__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [S: set(A),F2: fun(A,fun(B,C)),Net: filter(B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => differentiable(B,C,aa(A,fun(B,C),F2,X3),Net) )
           => differentiable(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ue(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2),Net) ) ) ) ).

% differentiable_sum
tff(fact_5194_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),top_top(A))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),top_top(A))
                & ! [Z6: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Z6)
                   => aa(A,$o,P,Z6) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_5195_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A)] :
          ( ! [X3: A,Y3: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P,X3)
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P,X3)
                 => aa(A,$o,Q,aa(B,A,G,X3)) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_5196_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,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,P,Y4) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_5197_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xb: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( eventually(A,P,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,P,Y4) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_5198_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),G: fun(A,B),Net: filter(A),H: fun(A,B),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_uf(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_uf(fun(A,B),fun(fun(A,B),fun(A,$o)),G),H),Net)
           => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),Net)
             => ( filterlim(A,B,H,topolo7230453075368039082e_nhds(B,C2),Net)
               => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_5199_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ug(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_5200_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_5201_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [A4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ui(fun(B,A),fun(A,fun(B,$o)),F2),A4),F4) )
         => ( ! [A4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A4)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_uj(fun(B,A),fun(A,fun(B,$o)),F2),A4),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F4) ) ) ) ).

% order_tendstoI
tff(fact_5202_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),Xb)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,$o)),F2),L4),F4) )
            & ! [U6: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xb),U6)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ug(fun(A,B),fun(B,fun(A,$o)),F2),U6),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_5203_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_uk(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
           => filterlim(A,B,G,at_top(B),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_5204_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ul(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_5205_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ul(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F4) ) ) ).

% filterlim_at_top
tff(fact_5206_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_um(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_5207_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z7),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_un(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_5208_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_un(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F4) ) ) ).

% filterlim_at_bot
tff(fact_5209_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_uo(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_5210_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Xb: A] :
          ~ ! [A6: fun(nat,set(A))] :
              ( ! [I3: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A6,I3))
             => ( ! [I3: nat] : member(A,Xb,aa(nat,set(A),A6,I3))
               => ~ ! [S8: set(A)] :
                      ( topolo1002775350975398744n_open(A,S8)
                     => ( member(A,Xb,S8)
                       => eventually(nat,aa(set(A),fun(nat,$o),aTP_Lamp_up(fun(nat,set(A)),fun(set(A),fun(nat,$o)),A6),S8),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_5211_eventually__Inf__base,axiom,
    ! [A: $tType,B3: set(filter(A)),P: fun(A,$o)] :
      ( ( B3 != bot_bot(set(filter(A))) )
     => ( ! [F5: filter(A)] :
            ( member(filter(A),F5,B3)
           => ! [G2: filter(A)] :
                ( member(filter(A),G2,B3)
               => ? [X: filter(A)] :
                    ( member(filter(A),X,B3)
                    & 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),G2)) ) ) )
       => ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3))
        <=> ? [X2: filter(A)] :
              ( member(filter(A),X2,B3)
              & eventually(A,P,X2) ) ) ) ) ).

% eventually_Inf_base
tff(fact_5212_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,$o),F4: fun(A,filter(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),A3)))
      <=> ? [Q7: fun(A,fun(B,$o))] :
            ( ! [X2: A] :
                ( member(A,X2,A3)
               => eventually(B,aa(A,fun(B,$o),Q7,X2),aa(A,filter(B),F4,X2)) )
            & ! [Y4: B] :
                ( ! [X2: A] :
                    ( member(A,X2,A3)
                   => aa(B,$o,aa(A,fun(B,$o),Q7,X2),Y4) )
               => aa(B,$o,P,Y4) ) ) ) ) ).

% eventually_INF_finite
tff(fact_5213_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,$o),aTP_Lamp_uq(real,fun(real,fun(real,$o)),B2),A2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ).

% eventually_at_left_real
tff(fact_5214_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ej(A,fun(fun(nat,A),fun(nat,A)),C2),F2),at_top(nat))
          <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_5215_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_5216_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,B,F2,Xb) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_ur(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% differentiable_inverse
tff(fact_5217_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A,S2: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X2: A] :
                  ( member(A,X2,S2)
                 => ( ( ( X2 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),D6) )
                   => aa(A,$o,P,X2) ) ) ) ) ) ).

% eventually_at
tff(fact_5218_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X2: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),D6)
                 => aa(A,$o,P,X2) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_5219_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X3: A] :
              ( member(A,X3,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_5220_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_us(fun(A,$o),fun(A,fun(A,$o)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_5221_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_ut(fun(A,B),fun(B,fun(A,$o)),F2),L),F4)
         => ( ! [X3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,$o)),F2),X3),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_5222_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F4: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_uu(B,fun(fun(A,B),fun(A,$o)),L),F2),F4)
         => ( ! [X3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X3)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ug(fun(A,B),fun(B,fun(A,$o)),F2),X3),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_5223_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_uv(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_5224_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z7),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_uw(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_5225_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F4: filter(A),F2: fun(A,B),Xb: B,G: fun(A,B),Y: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Y),F4)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ux(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Xb) ) ) ) ) ) ).

% tendsto_le
tff(fact_5226_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_uy(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),Xb) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_5227_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_uz(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),A2) ) ) ) ) ).

% tendsto_upperbound
tff(fact_5228_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_va(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,F2,at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_5229_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_vb(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,F2,at_bot(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_5230_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F4: fun(B,filter(A)),B3: set(B)] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),B3)))
    <=> ? [X8: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X8),B3)
          & aa(set(B),$o,finite_finite2(B),X8)
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F4),X8))) ) ) ).

% eventually_INF
tff(fact_5231_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A,S2: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X2: A] :
                  ( member(A,X2,S2)
                 => ( ( ( X2 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X2,A2)),D6) )
                   => aa(A,$o,P,X2) ) ) ) ) ) ).

% eventually_at_le
tff(fact_5232_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P3: fun(A,$o)] :
          ( eventually(A,P3,at_infinity(A))
        <=> ? [B7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B7)
              & ! [X2: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X2))
                 => aa(A,$o,P3,X2) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_5233_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_ug(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_lessThan(B),L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_5234_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),E2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( aa(real,$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,$o)),aTP_Lamp_vc(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E2),F4) ) ) ) ).

% tendstoD
tff(fact_5235_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_vc(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E),F4) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% tendstoI
tff(fact_5236_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E3: real] :
              ( aa(real,$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,$o)),aTP_Lamp_vc(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E3),F4) ) ) ) ).

% tendsto_iff
tff(fact_5237_eventually__Inf,axiom,
    ! [A: $tType,P: fun(A,$o),B3: set(filter(A))] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3))
    <=> ? [X8: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X8),B3)
          & aa(set(filter(A)),$o,finite_finite2(filter(A)),X8)
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X8)) ) ) ).

% eventually_Inf
tff(fact_5238_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_5239_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_5240_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K5: real,X5: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
         => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K5)
           => bfun(nat,A,X5,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_5241_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,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N2))),K6) ) ) ) ).

% Bseq_def
tff(fact_5242_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [N6: nat] :
            ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% Bseq_iff1a
tff(fact_5243_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A2: A] :
          ( ! [X3: A,Y3: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P,X3)
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P,X3)
                 => aa(A,$o,Q,aa(B,A,G,X3)) )
             => ( 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,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_5244_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F4: fun(A,filter(B)),P: fun(B,$o)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( member(A,A4,B3)
           => ! [B4: A] :
                ( member(A,B4,B3)
               => ? [X: A] :
                    ( member(A,X,B3)
                    & 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,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F4),B3)))
        <=> ? [X2: A] :
              ( member(A,X2,B3)
              & eventually(B,P,aa(A,filter(B),F4,X2)) ) ) ) ) ).

% eventually_INF_base
tff(fact_5245_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,C),K5: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_vd(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K5),F4)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_5246_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),A3: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ve(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),C2),A3),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,A3),F4) ) ) ) ).

% filterlim_at_withinI
tff(fact_5247_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C2: real,F2: fun(A,B),F4: filter(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
         => ( filterlim(A,B,F2,at_infinity(B),F4)
          <=> ! [R5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),R5)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_vf(fun(A,B),fun(real,fun(A,$o)),F2),R5),F4) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_5248_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_vg(fun(A,real),fun(A,$o),F2),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_vh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_5249_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_vg(fun(A,real),fun(A,$o),F2),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_vh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ) ).

% tendsto_powr2
tff(fact_5250_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( ( ( A2 != zero_zero(real) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
              & eventually(A,aTP_Lamp_vg(fun(A,real),fun(A,$o),F2),F4) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ).

% tendsto_powr'
tff(fact_5251_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => ( eventually(A,aTP_Lamp_va(fun(A,real),fun(A,$o),G),F4)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vi(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_5252_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_va(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),F2),at_top(real),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_5253_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_va(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ) ).

% filterlim_inverse_at_top
tff(fact_5254_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_vb(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ) ).

% filterlim_inverse_at_bot
tff(fact_5255_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
              & ? [N6: nat] :
                ! [N2: nat] : aa(real,$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,N2)),aa(A,A,uminus_uminus(A),aa(nat,A,X5,N6))))),K2) ) ) ) ).

% Bseq_iff3
tff(fact_5256_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
              & ? [X2: A] :
                ! [N2: nat] : aa(real,$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,N2)),aa(A,A,uminus_uminus(A),X2)))),K2) ) ) ) ).

% Bseq_iff2
tff(fact_5257_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_vk(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_Cauchy'
tff(fact_5258_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F4: filter(A)] :
          ( topolo6773858410816713723filter(A,F4)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [P5: fun(A,$o)] :
                  ( eventually(A,P5,F4)
                  & ! [X2: A,Y4: A] :
                      ( ( aa(A,$o,P5,X2)
                        & aa(A,$o,P5,Y4) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Y4)),E3) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_5259_MVT,axiom,
    ! [A2: real,B2: real,F2: 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),F2)
       => ( ! [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,F2,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,F2,L3,topolo174197925503356063within(real,Z2,top_top(set(real))))
              & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),L3) ) ) ) ) ) ).

% MVT
tff(fact_5260_continuous__on__sing,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,F2: fun(A,B)] : topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))),F2) ) ).

% continuous_on_sing
tff(fact_5261_continuous__on__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(A),F2: fun(A,B),Ta: set(A)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S)
           => topolo81223032696312382ous_on(A,B,Ta,F2) ) ) ) ).

% continuous_on_subset
tff(fact_5262_continuous__on__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_vl(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_add
tff(fact_5263_IVT2_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),B2: B,Y: A,A2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,B2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,A2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A2,B2),F2)
               => ? [X3: B] :
                    ( aa(B,$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,F2,X3) = Y ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_5264_IVT_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),A2: B,Y: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,A2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A2,B2),F2)
               => ? [X3: B] :
                    ( aa(B,$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,F2,X3) = Y ) ) ) ) ) ) ) ).

% IVT'
tff(fact_5265_continuous__on__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B)] : topolo81223032696312382ous_on(A,B,bot_bot(set(A)),F2) ) ).

% continuous_on_empty
tff(fact_5266_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => ( aa(A,B,G,X3) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_vm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_5267_continuous__on__compose2,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Ta: set(A),G: fun(A,B),S: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(A,B,Ta,G)
         => ( topolo81223032696312382ous_on(C,A,S,F2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),S)),Ta)
             => topolo81223032696312382ous_on(C,B,S,aa(fun(C,A),fun(C,B),aTP_Lamp_vn(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2)) ) ) ) ) ).

% continuous_on_compose2
tff(fact_5268_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_vo(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_5269_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aTP_Lamp_vp(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_5270_continuous__onI__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & dense_order(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(B,A),A3: set(B)] :
          ( topolo1002775350975398744n_open(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( ! [X3: B,Y3: B] :
                ( member(B,X3,A3)
               => ( member(B,Y3,A3)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y3)) ) ) )
           => topolo81223032696312382ous_on(B,A,A3,F2) ) ) ) ).

% continuous_onI_mono
tff(fact_5271_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_vq(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_5272_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( cos(A,aa(A,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_ql(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_5273_open__Collect__less__Int,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ? [A6: set(A)] :
                ( topolo1002775350975398744n_open(A,A6)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),S) = collect(A,aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_vr(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),S),F2),G)) ) ) ) ) ) ).

% open_Collect_less_Int
tff(fact_5274_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_vs(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_5275_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( sin(A,aa(A,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S,aTP_Lamp_qj(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_5276_continuous__on__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => ( cosh(B,aa(A,B,F2,X3)) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,A3,aTP_Lamp_vt(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_5277_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),D2: B] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => aa(set(fun(A,B)),$o,finite_finite2(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_vu(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A3),B3),D2))) ) ) ).

% finite_set_of_finite_funs
tff(fact_5278_open__Collect__positive,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ? [A6: set(A)] :
              ( topolo1002775350975398744n_open(A,A6)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),S) = collect(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_vv(set(A),fun(fun(A,real),fun(A,$o)),S),F2)) ) ) ) ) ).

% open_Collect_positive
tff(fact_5279_continuous__on__powr_H,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F2,X3))
                    & ( ( aa(A,real,F2,X3) = zero_zero(real) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X3)) ) ) )
             => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_vw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_5280_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F2)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,X3)) )
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ( aa(A,real,F2,X3) != one_one(real) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,S)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X3)) )
                 => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_vx(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_5281_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,B2: A,F2: fun(A,A)] :
          ( ! [X3: A] :
              ( aa(A,$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)
               => ? [Y2: A] : has_field_derivative(A,F2,Y2,topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_5282_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F2: 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,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [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,F2,aa(real,fun(real,real),F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),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_5283_mvt,axiom,
    ! [A2: real,B2: real,F2: 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),F2)
       => ( ! [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,F2,aa(real,fun(real,real),F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),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,F2,B2)),aa(real,real,F2,A2)) != aa(real,real,aa(real,fun(real,real),F9,Xi),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_5284_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_5285_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F2: 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),F2)
       => ( ! [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,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B2) = aa(real,real,F2,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_5286_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F2: 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)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),zero_zero(real)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_5287_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F2: 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)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_5288_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F2: 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),F2)
       => ( ! [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,F2,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,F2,Xb) = aa(real,real,F2,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_5289_Rolle,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ( aa(real,real,F2,A2) = aa(real,real,F2,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => ( ! [X3: real] :
                ( 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,F2,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,F2,zero_zero(real),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_5290_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_vy(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_5291_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_vz(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_5292_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),P: fun(A,$o)] : aa(fun(A,$o),A,least(A,Less_eq),P) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_wa(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),P)) ).

% ord.Least_def
tff(fact_5293_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or3652927894154168847AtMost(nat,L,U)) ).

% finite_greaterThanAtMost
tff(fact_5294_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,L: A,U: A] :
          ( member(A,Ia,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Ia)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ia),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_5295_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_5296_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_5297_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_5298_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_5299_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_or3652927894154168847AtMost(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ).

% image_add_greaterThanAtMost
tff(fact_5300_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,Xb)) = Xb ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_5301_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Xb,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_5302_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,Xb)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_5303_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Xb,Y)) = Xb ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_5304_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_5305_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or7035219750837199246ssThan(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ).

% image_diff_atLeastLessThan
tff(fact_5306_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or3652927894154168847AtMost(A,A2,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_5307_image__uminus__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or7035219750837199246ssThan(A,Xb,Y)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_atLeastLessThan
tff(fact_5308_image__uminus__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or3652927894154168847AtMost(A,Xb,Y)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),Xb)) ) ).

% image_uminus_greaterThanAtMost
tff(fact_5309_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C2,D2) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2) )
            | ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_5310_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_5311_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_5312_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_5313_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_5314_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_5315_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B2: nat] :
      ( ? [X_12: nat] : aa(nat,$o,P,X_12)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_5316_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_5317_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_5318_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),A2) ) ) ) ).

% Ioc_disjoint
tff(fact_5319_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),Xb: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( member(A,Xb,S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
             => ? [B4: A] :
                  ( aa(A,$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_5320_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_5321_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_5322_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_5323_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
           => ( order_Greatest(A,P) = Xb ) ) ) ) ).

% Greatest_equality
tff(fact_5324_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
           => ( ! [X3: A] :
                  ( aa(A,$o,P,X3)
                 => ( ! [Y2: A] :
                        ( aa(A,$o,P,Y2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X3) )
                   => aa(A,$o,Q,X3) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_5325_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.head
tff(fact_5326_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.head
tff(fact_5327_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_5328_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_5329_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_5330_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or3652927894154168847AtMost(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_5331_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,Ia: 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),Ia))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,Ia,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5332_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,B2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2)))
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),B2)
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X3)),topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_5333_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,$o),F2: fun(A,B),A3: B,B3: set(B)] :
          ( eventually(A,P,filtercomap(A,B,F2,topolo174197925503356063within(B,A3,B3)))
        <=> ? [S9: set(B)] :
              ( topolo1002775350975398744n_open(B,S9)
              & member(B,A3,S9)
              & ! [X2: A] :
                  ( member(B,aa(A,B,F2,X2),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),B3)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))))
                 => aa(A,$o,P,X2) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_5334_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_tx(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_5335_greaterThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,set(A),set_ord_greaterThan(A),Xb) = aa(A,set(A),set_ord_greaterThan(A),Y) )
        <=> ( Xb = Y ) ) ) ).

% greaterThan_eq_iff
tff(fact_5336_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_5337_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,K: A] :
          ( member(A,Ia,aa(A,set(A),set_ord_greaterThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),Ia) ) ) ).

% greaterThan_iff
tff(fact_5338_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_5339_filtercomap__bot,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : filtercomap(A,B,F2,bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtercomap_bot
tff(fact_5340_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),Xb)),aa(A,set(A),set_ord_greaterThan(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% greaterThan_subset_iff
tff(fact_5341_Compl__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_greaterThan(A),K)) = aa(A,set(A),set_ord_atMost(A),K) ) ).

% Compl_greaterThan
tff(fact_5342_Compl__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atMost(A),K)) = aa(A,set(A),set_ord_greaterThan(A),K) ) ).

% Compl_atMost
tff(fact_5343_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_5344_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_5345_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_5346_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_5347_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_5348_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_5349_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = collect(A,aa(A,fun(A,$o),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_5350_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_5351_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_5352_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_5353_filtercomap__neq__bot,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F2: fun(B,A)] :
      ( ! [P6: fun(A,$o)] :
          ( eventually(A,P6,F4)
         => ? [X: B] : aa(A,$o,P6,aa(B,A,F2,X)) )
     => ( filtercomap(B,A,F2,F4) != bot_bot(filter(B)) ) ) ).

% filtercomap_neq_bot
tff(fact_5354_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,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,P,Y4) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_5355_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,Y: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( eventually(A,P,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,P,Y4) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_5356_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_5357_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_5358_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_5359_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_5360_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_5361_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_5362_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_5363_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_5364_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_5365_eventually__filtercomap__at__top__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_top(B)))
        <=> ? [N6: B] :
            ! [X2: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N6),aa(A,B,F2,X2))
             => aa(A,$o,P,X2) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_5366_eventually__filtercomap__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_top(B)))
        <=> ? [N6: B] :
            ! [X2: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N6),aa(A,B,F2,X2))
             => aa(A,$o,P,X2) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_5367_filtercomap__neq__bot__surj,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),F2: fun(B,A)] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
       => ( filtercomap(B,A,F2,F4) != bot_bot(filter(B)) ) ) ) ).

% filtercomap_neq_bot_surj
tff(fact_5368_eventually__filtercomap__at__bot__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N6: B] :
            ! [X2: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),N6)
             => aa(A,$o,P,X2) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_5369_eventually__filtercomap__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N6: B] :
            ! [X2: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),N6)
             => aa(A,$o,P,X2) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_5370_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [A4: A,B4: A] :
              ( member(A,Xb,aa(A,set(A),set_ord_lessThan(A),A4))
              & member(A,Y,aa(A,set(A),set_ord_greaterThan(A),B4))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),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_5371_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X3: A] :
              ( member(A,X3,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% eventually_at_rightI
tff(fact_5372_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_5373_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,$o),aTP_Lamp_uq(real,fun(real,fun(real,$o)),A2),B2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ).

% eventually_at_right_real
tff(fact_5374_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_5375_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P3: B,F12: filter(A),C2: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P3,aa(B,set(B),set_ord_greaterThan(B),P3)),F12)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => ( ( L = aa(B,B,aa(B,fun(B,B),times_times(B),C2),P3) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wb(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo174197925503356063within(B,L,aa(B,set(B),set_ord_greaterThan(B),L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_5376_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_greaterThan(B),L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_5377_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_5378_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A2: A] :
          ( ! [X3: A,Y3: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P,X3)
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P,X3)
                 => aa(A,$o,Q,aa(B,A,G,X3)) )
             => ( 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,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_5379_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_5380_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_wc(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_5381_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)),insert2(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_wd(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_we(A,fun(set(A),fun(A,filter(A))),Xb),S)),aa(A,set(A),set_ord_lessThan(A),Xb)))) ) ) ) ).

% at_within_order
tff(fact_5382_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_5383_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,F3,N4))
               => ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N4)),B2)
                 => ( order_antimono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_wf(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_5384_atLeast__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,set(A),set_ord_atLeast(A),Xb) = aa(A,set(A),set_ord_atLeast(A),Y) )
        <=> ( Xb = Y ) ) ) ).

% atLeast_eq_iff
tff(fact_5385_atLeast__0,axiom,
    aa(nat,set(nat),set_ord_atLeast(nat),zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_5386_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Ia: A,K: A] :
          ( member(A,Ia,aa(A,set(A),set_ord_atLeast(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),Ia) ) ) ).

% atLeast_iff
tff(fact_5387_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_5388_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_5389_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),Xb)),aa(A,set(A),set_ord_atLeast(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% atLeast_subset_iff
tff(fact_5390_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,Ia: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),aa(A,set(A),set_ord_atLeast(A),Ia)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K),Ia)) ) ).

% image_add_atLeast
tff(fact_5391_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_5392_principal__le__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),principal(A,A3)),principal(A,B3))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ).

% principal_le_iff
tff(fact_5393_Compl__atLeast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atLeast(A),K)) = aa(A,set(A),set_ord_lessThan(A),K) ) ).

% Compl_atLeast
tff(fact_5394_Compl__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_lessThan(A),K)) = aa(A,set(A),set_ord_atLeast(A),K) ) ).

% Compl_lessThan
tff(fact_5395_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_5396_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),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),C2),A2)) ) ).

% image_minus_const_atLeast
tff(fact_5397_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),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),C2),B2)) ) ).

% image_minus_const_AtMost
tff(fact_5398_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_5399_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_5400_Int__atLeastAtMostR2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,D2: 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,C2,D2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),D2) ) ).

% Int_atLeastAtMostR2
tff(fact_5401_Int__atLeastAtMostL2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: 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),C2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),B2) ) ).

% Int_atLeastAtMostL2
tff(fact_5402_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_greaterThan(nat),K) ).

% atLeast_Suc_greaterThan
tff(fact_5403_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,Xb)) ) ) ) ).

% antimonoD
tff(fact_5404_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,Xb)) ) ) ) ).

% antimonoE
tff(fact_5405_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y3)),aa(A,B,F2,X3)) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_5406_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X2: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y4)),aa(A,B,F2,X2)) ) ) ) ).

% antimono_def
tff(fact_5407_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = collect(A,aa(A,fun(A,$o),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_5408_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_5409_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_5410_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_5411_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_5412_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_5413_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_5414_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_5415_bot__eq__principal__empty,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = principal(A,bot_bot(set(A))) ).

% bot_eq_principal_empty
tff(fact_5416_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_5417_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_5418_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_5419_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_5420_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_5421_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),Ia: nat] :
          ( order_antimono(nat,A,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,aa(nat,nat,suc,Ia))),aa(nat,A,A3,Ia)) ) ) ).

% decseq_SucD
tff(fact_5422_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N))),aa(nat,A,X5,N))
         => order_antimono(nat,A,X5) ) ) ).

% decseq_SucI
tff(fact_5423_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N2))),aa(nat,A,F2,N2)) ) ) ).

% decseq_Suc_iff
tff(fact_5424_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( order_antimono(nat,A,X5)
        <=> ! [M3: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,M3)) ) ) ) ).

% decseq_def
tff(fact_5425_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Ia: nat,J: nat] :
          ( order_antimono(nat,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,J)),aa(nat,A,F2,Ia)) ) ) ) ).

% decseqD
tff(fact_5426_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)),insert2(A),Xb),bot_bot(set(A)))) ) ).

% nhds_discrete
tff(fact_5427_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_5428_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_5429_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_5430_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_5431_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_5432_tendsto__principal__singleton,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),Xb: A] : filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xb)),principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) ) ).

% tendsto_principal_singleton
tff(fact_5433_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)),insert2(A),Xb),bot_bot(set(A))))
         => ( topolo7230453075368039082e_nhds(A,Xb) = principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) ) ) ) ).

% nhds_discrete_open
tff(fact_5434_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)),insert2(A),Nb),bot_bot(set(A))) ) ).

% atMost_Int_atLeast
tff(fact_5435_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_5436_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_5437_atLeast__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atLeast(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_5438_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_wh(A,fun(real,filter(A)),Xb)),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% nhds_metric
tff(fact_5439_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F4: fun(A,set(B)),F2: fun(B,C),G3: fun(D,set(C)),J4: set(D)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => ! [J2: A] :
                ( member(A,J2,I5)
               => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,I2)),aa(A,set(B),F4,J2))
                  | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,J2)),aa(A,set(B),F4,I2)) ) ) )
       => ( filterlim(B,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_wi(fun(D,set(C)),fun(D,filter(C)),G3)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_wj(fun(A,set(B)),fun(A,filter(B)),F4)),I5)))
        <=> ! [X2: D] :
              ( member(D,X2,J4)
             => ? [Xa3: A] :
                  ( member(A,Xa3,I5)
                  & ! [Xb4: B] :
                      ( member(B,Xb4,aa(A,set(B),F4,Xa3))
                     => member(C,aa(B,C,F2,Xb4),aa(D,set(C),G3,X2)) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_5440_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F2: fun(A,set(B))] :
      ( aa(set(A),$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_wj(fun(A,set(B)),fun(A,filter(B)),F2)),X5)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),X5))) ) ) ).

% INF_principal_finite
tff(fact_5441_at__within__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S: set(A)] : topolo174197925503356063within(A,A2,S) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_5442_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),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_wk(A,fun(A,filter(A)),Xb)),aa(A,set(A),set_ord_lessThan(A),Xb))) ) ) ) ).

% at_left_eq
tff(fact_5443_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( 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_wl(A,fun(A,filter(A)),Xb)),aa(A,set(A),set_ord_greaterThan(A),Xb))) ) ) ) ).

% at_right_eq
tff(fact_5444_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A,S: set(A)] : topolo174197925503356063within(A,Xb,S) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image2(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_wm(A,fun(set(A),fun(set(A),filter(A))),Xb),S)),collect(set(A),aTP_Lamp_wn(A,fun(set(A),$o),Xb)))) ) ).

% at_within_eq
tff(fact_5445_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,$o),ord_less(A),A2),B2)
         => ( ! [S5: fun(nat,A)] :
                ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,S5,N4))
               => ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S5,N4)),B2)
                 => ( order_antimono(nat,A,S5)
                   => ( filterlim(nat,A,S5,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wo(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S5),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_5446_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),$o),ord_less_eq(filter(A)),F10),principal(A,S2))
             => ( ( F10 != bot_bot(filter(A)) )
               => ( topolo6773858410816713723filter(A,F10)
                 => ? [X2: A] :
                      ( member(A,X2,S2)
                      & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F10),topolo7230453075368039082e_nhds(A,X2)) ) ) ) ) ) ) ).

% complete_uniform
tff(fact_5447_continuous__at__Sup__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( order_antimono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S2),aa(A,set(A),set_ord_lessThan(A),aa(set(A),A,complete_Sup_Sup(A),S2))),F2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S2)
               => ( aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),S2)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),S2)) ) ) ) ) ) ) ).

% continuous_at_Sup_antimono
tff(fact_5448_continuous__at__Inf__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( order_antimono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Inf_Inf(A),S2),aa(A,set(A),set_ord_greaterThan(A),aa(set(A),A,complete_Inf_Inf(A),S2))),F2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S2)
               => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),S2)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),S2)) ) ) ) ) ) ) ).

% continuous_at_Inf_antimono
tff(fact_5449_bdd__below_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),M4: A] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M4),X3) )
         => condit1013018076250108175_below(A,A3) ) ) ).

% bdd_below.I
tff(fact_5450_bdd__belowI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),Mb: A] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X3) )
         => condit1013018076250108175_below(A,A3) ) ) ).

% bdd_belowI
tff(fact_5451_bdd__above_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),M4: A] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),M4) )
         => condit941137186595557371_above(A,A3) ) ) ).

% bdd_above.I
tff(fact_5452_bdd__below__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit1013018076250108175_below(A,bot_bot(set(A))) ) ).

% bdd_below_empty
tff(fact_5453_bdd__above__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit941137186595557371_above(A,bot_bot(set(A))) ) ).

% bdd_above_empty
tff(fact_5454_bdd__below__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I5: set(A),A3: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( condit1013018076250108175_below(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5)))
          <=> ! [X2: A] :
                ( member(A,X2,I5)
               => condit1013018076250108175_below(B,aa(A,set(B),A3,X2)) ) ) ) ) ).

% bdd_below_UN
tff(fact_5455_bdd__above__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I5: set(A),A3: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5)))
          <=> ! [X2: A] :
                ( member(A,X2,I5)
               => condit941137186595557371_above(B,aa(A,set(B),A3,X2)) ) ) ) ) ).

% bdd_above_UN
tff(fact_5456_bdd__below__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => condit1013018076250108175_below(A,A3) ) ) ).

% bdd_below_finite
tff(fact_5457_bdd__above__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => condit941137186595557371_above(A,A3) ) ) ).

% bdd_above_finite
tff(fact_5458_bdd__above__nat,axiom,
    ! [X5: set(nat)] :
      ( condit941137186595557371_above(nat,X5)
    <=> aa(set(nat),$o,finite_finite2(nat),X5) ) ).

% bdd_above_nat
tff(fact_5459_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A3)
           => ( condit1013018076250108175_below(A,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ) ).

% cInf_le_cSup
tff(fact_5460_bdd__below_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit1013018076250108175_below(A,A3)
         => ~ ! [M7: A] :
                ~ ! [X: A] :
                    ( member(A,X,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M7),X) ) ) ) ).

% bdd_below.E
tff(fact_5461_bdd__below_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit1013018076250108175_below(A,A3)
        <=> ? [M8: A] :
            ! [X2: A] :
              ( member(A,X2,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M8),X2) ) ) ) ).

% bdd_below.unfold
tff(fact_5462_bdd__above_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit941137186595557371_above(A,A3)
         => ~ ! [M7: A] :
                ~ ! [X: A] :
                    ( member(A,X,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M7) ) ) ) ).

% bdd_above.E
tff(fact_5463_bdd__above_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit941137186595557371_above(A,A3)
        <=> ? [M8: A] :
            ! [X2: A] :
              ( member(A,X2,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),M8) ) ) ) ).

% bdd_above.unfold
tff(fact_5464_bdd__above__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: set(A),A3: set(A)] :
          ( condit941137186595557371_above(A,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => condit941137186595557371_above(A,A3) ) ) ) ).

% bdd_above_mono
tff(fact_5465_bdd__below__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: set(A),A3: set(A)] :
          ( condit1013018076250108175_below(A,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => condit1013018076250108175_below(A,A3) ) ) ) ).

% bdd_below_mono
tff(fact_5466_bdd__below_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A3: set(A),M4: B,F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M4),aa(A,B,F2,X3)) )
         => condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ).

% bdd_below.I2
tff(fact_5467_bdd__belowI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A3: set(A),Mb: B,F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F2,X3)) )
         => condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ).

% bdd_belowI2
tff(fact_5468_bdd__above_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A3: set(A),F2: fun(A,B),M4: B] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),M4) )
         => condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ).

% bdd_above.I2
tff(fact_5469_cInf__lower2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A),Y: A] :
          ( member(A,Xb,X5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( condit1013018076250108175_below(A,X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Y) ) ) ) ) ).

% cInf_lower2
tff(fact_5470_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_5471_cSup__upper2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A),Y: A] :
          ( member(A,Xb,X5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
           => ( condit941137186595557371_above(A,X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ).

% cSup_upper2
tff(fact_5472_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_5473_cINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( member(B,Xb,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(B,A,F2,Xb)) ) ) ) ).

% cINF_lower
tff(fact_5474_cINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: B,U: A] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( member(B,Xb,A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,Xb)),U)
             => aa(A,$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,F2),A3))),U) ) ) ) ) ).

% cINF_lower2
tff(fact_5475_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( ( B3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A3)
           => ( ! [B4: A] :
                  ( member(A,B4,B3)
                 => ? [X: A] :
                      ( member(A,X,A3)
                      & aa(A,$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),A3)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ) ) ).

% cInf_mono
tff(fact_5476_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,$o),ord_less_eq(A),A2),aa(set(A),A,complete_Inf_Inf(A),S2))
            <=> ! [X2: A] :
                  ( member(A,X2,S2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X2) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_5477_cSUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Xb: A,A3: set(A),F2: fun(A,B)] :
          ( member(A,Xb,A3)
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ).

% cSUP_upper
tff(fact_5478_cSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: B,U: A] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( member(B,Xb,A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F2,Xb))
             => aa(A,$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,F2),A3))) ) ) ) ) ).

% cSUP_upper2
tff(fact_5479_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Y: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Y)
            <=> ? [X2: A] :
                  ( member(A,X2,X5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_5480_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( ( B3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A3)
           => ( ! [B4: A] :
                  ( member(A,B4,B3)
                 => ? [X: A] :
                      ( member(A,X,A3)
                      & aa(A,$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),B3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ) ).

% cSup_mono
tff(fact_5481_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,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S2)),A2)
            <=> ! [X2: A] :
                  ( member(A,X2,S2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),A2) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_5482_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Y: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
            <=> ? [X2: A] :
                  ( member(A,X2,X5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X2) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_5483_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,Ia: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A3)))
           => ( member(B,Ia,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,Ia)) ) ) ) ) ).

% less_cINF_D
tff(fact_5484_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,Ia: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A3))),Y)
           => ( member(B,Ia,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Ia)),Y) ) ) ) ) ).

% cSUP_lessD
tff(fact_5485_cINF__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B3: set(A),F2: fun(C,B),A3: set(C),G: fun(A,B)] :
          ( ( B3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F2),A3))
           => ( ! [M2: A] :
                  ( member(A,M2,B3)
                 => ? [X: C] :
                      ( member(C,X,A3)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F2,X)),aa(A,B,G,M2)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ) ).

% cINF_mono
tff(fact_5486_le__cINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3)))
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,X2)) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_5487_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,B3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B3)),aa(set(A),A,complete_Inf_Inf(A),A3)) ) ) ) ) ).

% cInf_superset_mono
tff(fact_5488_cSUP__mono,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),G: fun(C,B),B3: set(C),F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,G),B3))
           => ( ! [N: A] :
                  ( member(A,N,A3)
                 => ? [X: C] :
                      ( member(C,X,B3)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,N)),aa(C,B,G,X)) ) )
             => aa(B,$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,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,G),B3))) ) ) ) ) ).

% cSUP_mono
tff(fact_5489_cSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),U)
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),U) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_5490_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B3)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ) ) ).

% cSup_subset_mono
tff(fact_5491_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)),insert2(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_5492_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)),insert2(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_5493_cINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),A2)
            <=> ? [X2: A] :
                  ( member(A,X2,A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),A2) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_5494_less__cSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3)))
            <=> ? [X2: A] :
                  ( member(A,X2,A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(A,B,F2,X2)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_5495_cINF__inf__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),A3))
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),A3))) = 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_wp(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A3)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_5496_cINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A),F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),B3))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
             => ( ! [X3: A] :
                    ( member(A,X3,B3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G,X3)),aa(A,B,F2,X3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_5497_cSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),G: fun(A,B),B3: set(A),F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),B3))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
             => ( ! [X3: A] :
                    ( member(A,X3,A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_5498_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( condit1013018076250108175_below(A,A3)
         => ( condit1013018076250108175_below(A,B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) != bot_bot(set(A)) )
             => aa(A,$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),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_5499_cINF__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A2)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ).

% cINF_insert
tff(fact_5500_cINF__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A3: set(A),B3: fun(A,set(B)),F2: fun(B,C)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => ( aa(A,set(B),B3,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_wq(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B3),F2)),A3)))
             => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3)))) = 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_wr(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B3),F2)),A3)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_5501_cSUP__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A3: set(A),B3: fun(A,set(B)),F2: fun(B,C)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => ( aa(A,set(B),B3,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_wq(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B3),F2)),A3)))
             => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3)))) = 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_ws(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B3),F2)),A3)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_5502_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_wu(real,filter(product_prod(A,A)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ).

% uniformity_dist
tff(fact_5503_compactE__image,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),C5: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [T5: B] :
                ( member(B,T5,C5)
               => topolo1002775350975398744n_open(A,aa(B,set(A),F2,T5)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),C5)))
             => ~ ! [C9: set(B)] :
                    ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C9),C5)
                   => ( aa(set(B),$o,finite_finite2(B),C9)
                     => ~ 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),F2),C9))) ) ) ) ) ) ) ).

% compactE_image
tff(fact_5504_compact__eq__Heine__Borel,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
        <=> ! [C7: set(set(A))] :
              ( ( ! [X2: set(A)] :
                    ( member(set(A),X2,C7)
                   => topolo1002775350975398744n_open(A,X2) )
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C7)) )
             => ? [D7: set(set(A))] :
                  ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),D7),C7)
                  & aa(set(set(A)),$o,finite_finite2(set(A)),D7)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),D7)) ) ) ) ) ).

% compact_eq_Heine_Borel
tff(fact_5505_uniformity__bot,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ( topolo7806501430040627800ormity(A) != bot_bot(filter(product_prod(A,A))) ) ) ).

% uniformity_bot
tff(fact_5506_compact__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo2193935891317330818ompact(A,bot_bot(set(A))) ) ).

% compact_empty
tff(fact_5507_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_5508_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_5509_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F2)
             => ? [X3: A] :
                  ( member(A,X3,S)
                  & ! [Xa: A] :
                      ( member(A,Xa,S)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,X3)) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_5510_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F2)
             => ? [X3: A] :
                  ( member(A,X3,S)
                  & ! [Xa: A] :
                      ( member(A,Xa,S)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Xa)) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_5511_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] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N2)
                 => ! [M3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M3)
                     => aa(product_prod(A,A),$o,P5,aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,X5,N2)),aa(nat,A,X5,M3))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_5512_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_ww(real,filter(product_prod(real,real)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_real_def
tff(fact_5513_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_wy(real,filter(product_prod(complex,complex)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_complex_def
tff(fact_5514_totally__bounded__def,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
        <=> ! [E5: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
             => ? [X8: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),X8)
                  & ! [X2: A] :
                      ( member(A,X2,S2)
                     => ? [Xa3: A] :
                          ( member(A,Xa3,X8)
                          & aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X2)) ) ) ) ) ) ) ).

% totally_bounded_def
tff(fact_5515_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),P,topolo7806501430040627800ormity(A))
        <=> ? [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
              & ! [X2: A,Y4: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Y4)),E3)
                 => aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),product_Pair(A,A,X2),Y4)) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_5516_compactE,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),T10: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T10))
           => ( ! [B9: set(A)] :
                  ( member(set(A),B9,T10)
                 => topolo1002775350975398744n_open(A,B9) )
             => ~ ! [T11: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),T11),T10)
                   => ( aa(set(set(A)),$o,finite_finite2(set(A)),T11)
                     => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T11)) ) ) ) ) ) ) ).

% compactE
tff(fact_5517_compactI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( ! [C8: set(set(A))] :
              ( ! [X: set(A)] :
                  ( member(set(A),X,C8)
                 => topolo1002775350975398744n_open(A,X) )
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C8))
               => ? [C10: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C10),C8)
                    & aa(set(set(A)),$o,finite_finite2(set(A)),C10)
                    & 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)),C10)) ) ) )
         => topolo2193935891317330818ompact(A,S) ) ) ).

% compactI
tff(fact_5518_prod__filter__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,I5: set(A),J4: set(B),A3: fun(A,filter(C)),B3: fun(B,filter(D))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ( J4 != bot_bot(set(B)) )
       => ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),A3),I5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B3),J4))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image2(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_xa(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J4),A3),B3)),I5)) ) ) ) ).

% prod_filter_INF
tff(fact_5519_prod__filter__INF1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A3: fun(A,filter(B)),B3: filter(C)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),A3),I5)),B3) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_xb(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A3),B3)),I5)) ) ) ).

% prod_filter_INF1
tff(fact_5520_prod__filter__INF2,axiom,
    ! [C: $tType,B: $tType,A: $tType,J4: set(A),A3: filter(B),B3: fun(A,filter(C))] :
      ( ( J4 != bot_bot(set(A)) )
     => ( prod_filter(B,C,A3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B3),J4))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_xc(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A3),B3)),J4)) ) ) ).

% prod_filter_INF2
tff(fact_5521_prod__filter__eq__bot,axiom,
    ! [A: $tType,B: $tType,A3: filter(A),B3: filter(B)] :
      ( ( prod_filter(A,B,A3,B3) = bot_bot(filter(product_prod(A,B))) )
    <=> ( ( A3 = bot_bot(filter(A)) )
        | ( B3 = bot_bot(filter(B)) ) ) ) ).

% prod_filter_eq_bot
tff(fact_5522_prod__filter__mono__iff,axiom,
    ! [A: $tType,B: $tType,A3: filter(A),B3: filter(B),C5: filter(A),D5: filter(B)] :
      ( ( A3 != bot_bot(filter(A)) )
     => ( ( B3 != bot_bot(filter(B)) )
       => ( aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,A3,B3)),prod_filter(A,B,C5,D5))
        <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),A3),C5)
            & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),B3),D5) ) ) ) ) ).

% prod_filter_mono_iff
tff(fact_5523_eventually__prod__sequentially,axiom,
    ! [P: fun(product_prod(nat,nat),$o)] :
      ( eventually(product_prod(nat,nat),P,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N6: nat] :
        ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M3)
         => ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N2)
             => aa(product_prod(nat,nat),$o,P,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,N2),M3)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_5524_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B3: filter(A),P: fun(B,$o),A3: filter(B)] :
      ( ( B3 != bot_bot(filter(A)) )
     => ( eventually(product_prod(B,A),product_case_prod(B,A,$o,aTP_Lamp_xd(fun(B,$o),fun(B,fun(A,$o)),P)),prod_filter(B,A,A3,B3))
      <=> eventually(B,P,A3) ) ) ).

% eventually_prod1
tff(fact_5525_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,A3: filter(A),P: fun(B,$o),B3: filter(B)] :
      ( ( A3 != bot_bot(filter(A)) )
     => ( eventually(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_xe(fun(B,$o),fun(A,fun(B,$o)),P)),prod_filter(A,B,A3,B3))
      <=> eventually(B,P,B3) ) ) ).

% eventually_prod2
tff(fact_5526_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: B,Xb: A,S: set(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),topolo174197925503356063within(A,Xb,S))
        <=> ! [X8: fun(nat,A)] :
              ( ! [I4: nat] : member(A,aa(nat,A,X8,I4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))
             => ( filterlim(nat,A,X8,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
               => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),comp(A,B,nat,F2),X8),topolo7230453075368039082e_nhds(B,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_5527_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,A2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,F3,N4))
               => ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N4)),A2)
                 => ( order_mono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_wf(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_5528_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_5529_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_xf(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_5530_funpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ).

% funpow_add
tff(fact_5531_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),A3: A,B3: A,Nb: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),A3)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),B3)) ) ) ) ).

% funpow_mono
tff(fact_5532_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( order_mono(nat,A,X5)
        <=> ! [M3: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M3)),aa(nat,A,X5,N2)) ) ) ) ).

% incseq_def
tff(fact_5533_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Ia: nat,J: nat] :
          ( order_mono(nat,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,Ia)),aa(nat,A,F2,J)) ) ) ) ).

% incseqD
tff(fact_5534_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),Ia: nat] :
          ( order_mono(nat,A,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,Ia)),aa(nat,A,A3,aa(nat,nat,suc,Ia))) ) ) ).

% incseq_SucD
tff(fact_5535_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,aa(nat,nat,suc,N)))
         => order_mono(nat,A,X5) ) ) ).

% incseq_SucI
tff(fact_5536_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_mono(nat,A,F2)
        <=> ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))) ) ) ).

% incseq_Suc_iff
tff(fact_5537_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_mono(A,B,F2)
        <=> ! [X2: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) ) ) ) ).

% mono_def
tff(fact_5538_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) )
         => order_mono(A,B,F2) ) ) ).

% monoI
tff(fact_5539_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% monoE
tff(fact_5540_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% monoD
tff(fact_5541_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mono_invE
tff(fact_5542_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),A3: A,B3: A] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A3)),aa(A,B,F2,B3))) ) ) ).

% mono_inf
tff(fact_5543_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% mono_strict_invE
tff(fact_5544_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_5545_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_5546_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,one_one(A),semiring_1_Nats(A)) ) ).

% Nats_1
tff(fact_5547_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_5548_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F2)
         => order_mono(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ) ) ).

% mono_pow
tff(fact_5549_comp__funpow,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F2: fun(B,B)] : aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aa(nat,fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),compow(fun(fun(A,B),fun(A,B))),Nb),comp(B,B,A,F2)) = comp(B,B,A,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Nb),F2)) ).

% comp_funpow
tff(fact_5550_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_5551_mono__Suc,axiom,
    order_mono(nat,nat,suc) ).

% mono_Suc
tff(fact_5552_comp__cong,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,E6: $tType,F2: fun(B,A),G: fun(C,B),Xb: C,F9: fun(D,A),G4: fun(E6,D),X4: E6] :
      ( ( aa(B,A,F2,aa(C,B,G,Xb)) = aa(D,A,F9,aa(E6,D,G4,X4)) )
     => ( aa(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F2),G),Xb) = aa(E6,A,aa(fun(E6,D),fun(E6,A),comp(D,A,E6,F9),G4),X4) ) ) ).

% comp_cong
tff(fact_5553_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_5554_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,zero_zero(A),semiring_1_Nats(A)) ) ).

% Nats_0
tff(fact_5555_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_5556_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: A,P: fun(A,$o)] :
          ( member(A,Xb,semiring_1_Nats(A))
         => ( ! [N: nat] : aa(A,$o,P,aa(nat,A,semiring_1_of_nat(A),N))
           => aa(A,$o,P,Xb) ) ) ) ).

% Nats_induct
tff(fact_5557_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: A] :
          ( member(A,Xb,semiring_1_Nats(A))
         => ~ ! [N: nat] : Xb != aa(nat,A,semiring_1_of_nat(A),N) ) ) ).

% Nats_cases
tff(fact_5558_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,F2),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ).

% funpow.simps(2)
tff(fact_5559_funpow__Suc__right,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)),F2) ).

% funpow_Suc_right
tff(fact_5560_cclfp__lowerbound,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,A),A3: A] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,A3)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),order_532582986084564980_cclfp(A,F2)),A3) ) ) ) ).

% cclfp_lowerbound
tff(fact_5561_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_5562_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_5563_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F2: fun(A,B),Mb: A,Nb: A,M6: B,N5: B] :
          ( order_mono(A,B,F2)
         => ( ( aa(set(A),set(B),image2(A,B,F2),set_or7035219750837199246ssThan(A,Mb,Nb)) = set_or7035219750837199246ssThan(B,M6,N5) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb)
             => ( aa(A,B,F2,Mb) = M6 ) ) ) ) ) ).

% mono_image_least
tff(fact_5564_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A3: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X3: B,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X3)),aa(B,A,H,Y3))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,H),G)),A3) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_5565_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F2: fun(A,A),P3: A,K: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,F2,P3))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_5566_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P3: A,K: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,P3)),P3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A))),P3) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_5567_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),Ia: nat,J: nat,Xb: A,Y: A] :
          ( order_mono(A,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,F2,Xb))
               => aa(A,$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)),Ia),F2),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),F2),Y)) ) ) ) ) ) ).

% funpow_mono2
tff(fact_5568_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,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,G),suc)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_5569_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,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,G),suc)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_5570_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_5571_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_5572_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,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,G),suc)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_5573_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,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,G),suc)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_5574_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_5575_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_5576_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_5577_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_5578_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xg(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A3),I5)))) ) ) ).

% mono_SUP
tff(fact_5579_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ).

% mono_Sup
tff(fact_5580_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xg(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))) ) ) ).

% mono_INF
tff(fact_5581_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ).

% mono_Inf
tff(fact_5582_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_xh(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_5583_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_5584_sum_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X3: A,Y3: A] :
                ( member(A,X3,A3)
               => ( member(A,Y3,A3)
                 => ( ( X3 != Y3 )
                   => ( ( aa(A,B,H,X3) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G,aa(A,B,H,X3)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(A),set(B),image2(A,B,H),A3)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H)),A3) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_5585_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Mb: nat,Nb: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( order_mono(A,A,F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2),top_top(A))) ) ) ) ).

% funpow_increasing
tff(fact_5586_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Mb: nat,Nb: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( order_mono(A,A,F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_5587_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X3: A,Y3: A] :
                ( member(A,X3,A3)
               => ( member(A,Y3,A3)
                 => ( ( X3 != Y3 )
                   => ( ( aa(A,B,H,X3) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G,aa(A,B,H,X3)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(A),set(B),image2(A,B,H),A3)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H)),A3) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_5588_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,F2,Y,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,F2),aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_5589_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(A,B,F2,aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_5590_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G,aa(A,C,F2,I2))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),aa(set(A),set(C),image2(A,C,F2),I5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,C),fun(A,B),comp(C,B,A,G),F2)),I5)) ) ) ) ).

% sum_image_le
tff(fact_5591_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D2: A,F4: filter(B),A2: A] :
          ( filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D2)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A))))
        <=> filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_5592_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A2: A,D2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D2)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_5593_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_5594_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_5595_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_5596_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_5597_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_5598_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),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,G),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_5599_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),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,G),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_5600_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F2)
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,A3),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xi(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A3),I5)))) ) ) ) ) ).

% mono_cSUP
tff(fact_5601_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( condit941137186595557371_above(A,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ) ).

% mono_cSup
tff(fact_5602_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F2)
         => ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,A3),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xi(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))) ) ) ) ) ).

% mono_cINF
tff(fact_5603_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( condit1013018076250108175_below(A,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ).

% mono_cInf
tff(fact_5604_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: 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,G),aTP_Lamp_xj(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),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_5605_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: 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,G),aTP_Lamp_xj(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),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_5606_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: 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,G),aTP_Lamp_xj(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),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_5607_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: 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,G),aTP_Lamp_xj(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),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_5608_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),Mb: nat,Nb: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),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,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_5609_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => order_mono(nat,nat,aTP_Lamp_xk(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_5610_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),Mb: nat,Nb: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),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,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_5611_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),Mb: nat,Nb: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),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,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_5612_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),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,G),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_5613_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$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),G),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,G),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_5614_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),Mb: nat,Nb: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),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,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_5615_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat))))
         => ( order_mono(nat,A,F2)
           => ( ! [N: nat] :
                  ( ( aa(nat,A,F2,N) = aa(nat,A,F2,aa(nat,nat,suc,N)) )
                 => ( aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,F2,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) )
             => ? [N8: nat] :
                  ( ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N8)
                     => ! [M: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N8)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N4)
                           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,M)),aa(nat,A,F2,N4)) ) ) )
                  & ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N4)
                     => ( aa(nat,A,F2,N8) = aa(nat,A,F2,N4) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_5616_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,$o),ord_less(A),B2),A2)
         => ( ! [S5: fun(nat,A)] :
                ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S5,N4)),A2)
               => ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,S5,N4))
                 => ( order_mono(nat,A,S5)
                   => ( filterlim(nat,A,S5,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wo(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S5),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_5617_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_5618_continuous__at__Sup__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( order_mono(A,B,F2)
         => ( 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))),F2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S2)
               => ( aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),S2)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),S2)) ) ) ) ) ) ) ).

% continuous_at_Sup_mono
tff(fact_5619_continuous__at__Inf__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( order_mono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Inf_Inf(A),S2),aa(A,set(A),set_ord_greaterThan(A),aa(set(A),A,complete_Inf_Inf(A),S2))),F2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S2)
               => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),S2)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),S2)) ) ) ) ) ) ) ).

% continuous_at_Inf_mono
tff(fact_5620_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y5: set(B),X5: set(A),F4: filter(B),F2: fun(A,C)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),Y5)),X5)
     => ( eventually(B,aTP_Lamp_xl(set(B),fun(B,$o),Y5),F4)
       => ( map_filter_on(A,C,X5,F2,map_filter_on(B,A,Y5,G,F4)) = map_filter_on(B,C,Y5,aa(fun(B,A),fun(B,C),comp(A,C,B,F2),G),F4) ) ) ) ).

% map_filter_on_comp
tff(fact_5621_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_5622_compact__imp__fip__image,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),I5: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [I2: B] :
                ( member(B,I2,I5)
               => topolo7761053866217962861closed(A,aa(B,set(A),F2,I2)) )
           => ( ! [I6: set(B)] :
                  ( aa(set(B),$o,finite_finite2(B),I6)
                 => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),I6),I5)
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),I6))) != bot_bot(set(A)) ) ) )
             => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),I5))) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip_image
tff(fact_5623_closed__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo7761053866217962861closed(A,bot_bot(set(A))) ) ).

% closed_empty
tff(fact_5624_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_5625_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_5626_closed__singleton,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [A2: A] : topolo7761053866217962861closed(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ).

% closed_singleton
tff(fact_5627_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_5628_closed__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A3: set(A),B3: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => topolo7761053866217962861closed(B,aa(A,set(B),B3,X3)) )
           => topolo7761053866217962861closed(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B3),A3))) ) ) ) ).

% closed_UN
tff(fact_5629_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F2: fun(A,C),G: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_xm(C,set(B))),F2) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image2(D,B,G)),aTP_Lamp_xn(A,set(D))) ).

% empty_natural
tff(fact_5630_mono__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A3: set(A),B3: set(A)] :
      ( order_mono(set(A),set(B),F2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F2,A3)),aa(set(A),set(B),F2,B3))) ) ).

% mono_Int
tff(fact_5631_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_5632_remdups__adj_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups_adj(A,nil(A)) = nil(A) ).

% remdups_adj.simps(1)
tff(fact_5633_remdups__adj__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups_adj(A,Xs) = Xs ) ) ).

% remdups_adj_distinct
tff(fact_5634_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_5635_closed__Collect__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo7761053866217962861closed(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_xo(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% closed_Collect_le
tff(fact_5636_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B3: set(set(A)),G: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,B3)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [A13: set(A)] :
                ( member(set(A),A13,B3)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B3)
                   => ( ( A13 != A24 )
                     => ! [X3: A] :
                          ( member(A,X3,A13)
                         => ( member(A,X3,A24)
                           => ( aa(A,B,G,X3) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),B3) ) ) ) ) ).

% prod.Union_comp
tff(fact_5637_t3__space,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [S2: set(A),Y: A] :
          ( topolo7761053866217962861closed(A,S2)
         => ( ~ member(A,Y,S2)
           => ? [U4: set(A),V5: set(A)] :
                ( topolo1002775350975398744n_open(A,U4)
                & topolo1002775350975398744n_open(A,V5)
                & member(A,Y,U4)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),V5)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),V5) = bot_bot(set(A)) ) ) ) ) ) ).

% t3_space
tff(fact_5638_t4__space,axiom,
    ! [A: $tType] :
      ( topological_t4_space(A)
     => ! [S2: set(A),T3: set(A)] :
          ( topolo7761053866217962861closed(A,S2)
         => ( topolo7761053866217962861closed(A,T3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3) = bot_bot(set(A)) )
             => ? [U4: set(A),V5: set(A)] :
                  ( topolo1002775350975398744n_open(A,U4)
                  & topolo1002775350975398744n_open(A,V5)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),U4)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),V5)
                  & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),V5) = bot_bot(set(A)) ) ) ) ) ) ) ).

% t4_space
tff(fact_5639_remdups__adj__adjacent,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ia)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),Ia) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,Ia)) ) ) ).

% remdups_adj_adjacent
tff(fact_5640_nhds__closed,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Xb: A,A3: set(A)] :
          ( member(A,Xb,A3)
         => ( topolo1002775350975398744n_open(A,A3)
           => ? [A10: set(A)] :
                ( member(A,Xb,A10)
                & topolo7761053866217962861closed(A,A10)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),A3)
                & eventually(A,aTP_Lamp_xp(set(A),fun(A,$o),A10),topolo7230453075368039082e_nhds(A,Xb)) ) ) ) ) ).

% nhds_closed
tff(fact_5641_continuous__on__closed__Union,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [I5: set(A),U2: fun(A,set(B)),F2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => topolo7761053866217962861closed(B,aa(A,set(B),U2,I2)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => topolo81223032696312382ous_on(B,C,aa(A,set(B),U2,I2),F2) )
             => topolo81223032696312382ous_on(B,C,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),U2),I5)),F2) ) ) ) ) ).

% continuous_on_closed_Union
tff(fact_5642_Lim__in__closed__set,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F2: fun(B,A),F4: filter(B),L: A] :
          ( topolo7761053866217962861closed(A,S2)
         => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_xq(set(A),fun(fun(B,A),fun(B,$o)),S2),F2),F4)
           => ( ( F4 != bot_bot(filter(B)) )
             => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
               => member(A,L,S2) ) ) ) ) ) ).

% Lim_in_closed_set
tff(fact_5643_prod_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(set(A)),G: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,C5)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [X3: set(A)] :
                ( member(set(A),X3,C5)
               => ! [Xa2: set(A)] :
                    ( member(set(A),Xa2,C5)
                   => ( ( X3 != Xa2 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa2) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),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)),G),C5) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_5644_sum_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B3: set(set(A)),G: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,B3)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [A13: set(A)] :
                ( member(set(A),A13,B3)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B3)
                   => ( ( A13 != A24 )
                     => ! [X3: A] :
                          ( member(A,X3,A13)
                         => ( member(A,X3,A24)
                           => ( aa(A,B,G,X3) = zero_zero(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G),B3) ) ) ) ) ).

% sum.Union_comp
tff(fact_5645_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_5646_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_5647_sum_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(set(A)),G: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,C5)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [X3: set(A)] :
                ( member(set(A),X3,C5)
               => ! [Xa2: set(A)] :
                    ( member(set(A),Xa2,C5)
                   => ( ( X3 != Xa2 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa2) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),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)),G),C5) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_5648_compact__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U2: set(A)] :
          ( topolo2193935891317330818ompact(A,U2)
        <=> ! [A8: set(set(A))] :
              ( ! [X2: set(A)] :
                  ( member(set(A),X2,A8)
                 => topolo7761053866217962861closed(A,X2) )
             => ( ! [B11: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),B11),A8)
                   => ( aa(set(set(A)),$o,finite_finite2(set(A)),B11)
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)) != bot_bot(set(A)) ) ) )
               => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A8)) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_fip
tff(fact_5649_compact__imp__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F4: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [T5: set(A)] :
                ( member(set(A),T5,F4)
               => topolo7761053866217962861closed(A,T5) )
           => ( ! [F11: set(set(A))] :
                  ( aa(set(set(A)),$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_5650_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_va(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,F2,at_top(real),F4)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_5651_Gcd__int__def,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),K5))) ).

% Gcd_int_def
tff(fact_5652_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_nl(nat,fun(real,real),Nb),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_5653_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5654_prod__encode__def,axiom,
    nat_prod_encode = product_case_prod(nat,nat,nat,aTP_Lamp_xs(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_5655_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : inj_on(A,B,F2,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_5656_remove1__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: A,F2: 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,F2),Xb),Xs)) = Xs ) ).

% remove1_insort_key
tff(fact_5657_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: 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,F2),Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_insort
tff(fact_5658_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_5659_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_xt(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_5660_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(A,A)),Xb),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5661_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3))
    <=> ( inj_on(A,B,F2,A3)
        & ~ member(B,aa(A,B,F2,A2),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_5662_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: 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_xr(A,A)),Y)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(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_xr(A,A)),Xb)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_5663_inj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( inj_on(A,A,F2,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A))) ) ).

% inj_fn
tff(fact_5664_card__image,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),nat,finite_card(A),A3) ) ) ).

% card_image
tff(fact_5665_inj__on__image__Fpow,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => inj_on(set(A),set(B),image2(A,B,F2),finite_Fpow(A,A3)) ) ).

% inj_on_image_Fpow
tff(fact_5666_finite__image__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,F2),A3))
      <=> aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% finite_image_iff
tff(fact_5667_finite__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
     => ( inj_on(B,A,F2,A3)
       => aa(set(B),$o,finite_finite2(B),A3) ) ) ).

% finite_imageD
tff(fact_5668_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(A),A3: set(A)] :
      ( inj_on(A,B,F2,B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B3)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3)) ) ) ).

% inj_on_strict_subset
tff(fact_5669_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3)
             => ( aa(A,B,F2,X3) != aa(A,B,F2,Y3) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_5670_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_xr(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_5671_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_xu(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_5672_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3) ) ).

% inj_on_add
tff(fact_5673_insort__key__left__comm,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xb: B,Y: B,Xs: list(B)] :
          ( ( aa(B,A,F2,Xb) != aa(B,A,F2,Y) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xb),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xb),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Y),Xs)) ) ) ) ).

% insort_key_left_comm
tff(fact_5674_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(A,A)),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(A,A)),Xb),Xs)) ) ).

% insort_left_comm
tff(fact_5675_insort__not__Nil,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: 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,F2),A2),Xs) != nil(A) ) ).

% insort_not_Nil
tff(fact_5676_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A),D5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( inj_on(B,A,F2,D5)
       => aa(set(B),$o,finite_finite2(B),collect(B,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_xv(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A3),F2),D5))) ) ) ).

% finite_inverse_image_gen
tff(fact_5677_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A3: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A3) ) ) ).

% inj_on_mult
tff(fact_5678_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_5679_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3)
             => ( member(A,X3,A3)
               => ( member(A,Y3,A3)
                 => ( aa(A,B,F2,X3) != aa(A,B,F2,Y3) ) ) ) )
         => ( ! [X3: A,Y3: A] :
                ( member(A,X3,A3)
               => ( member(A,Y3,A3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X3) ) ) )
           => inj_on(A,B,F2,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_5680_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => inj_on(A,B,F2,B3) ) ) ).

% inj_on_subset
tff(fact_5681_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(A),A3: set(A)] :
      ( inj_on(A,B,F2,B3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
       => inj_on(A,B,F2,A3) ) ) ).

% subset_inj_on
tff(fact_5682_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(B,A),T3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(B),set(A),image2(B,A,F2),T3))
    <=> ? [U5: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),U5),T3)
          & inj_on(B,A,F2,U5)
          & ( S2 = aa(set(B),set(A),image2(B,A,F2),U5) ) ) ) ).

% subset_image_inj
tff(fact_5683_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(A),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,B3)
     => ( member(A,A2,B3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ( member(B,aa(A,B,F2,A2),aa(set(A),set(B),image2(A,B,F2),A3))
          <=> member(A,A2,A3) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_5684_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C5: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
         => ( ( aa(set(A),set(B),image2(A,B,F2),A3) = aa(set(A),set(B),image2(A,B,F2),B3) )
          <=> ( A3 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_5685_inj__on__Inter,axiom,
    ! [B: $tType,A: $tType,S2: set(set(A)),F2: fun(A,B)] :
      ( ( S2 != bot_bot(set(set(A))) )
     => ( ! [A6: set(A)] :
            ( member(set(A),A6,S2)
           => inj_on(A,B,F2,A6) )
       => inj_on(A,B,F2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S2)) ) ) ).

% inj_on_Inter
tff(fact_5686_set__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: 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,F2),Xb),Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_key
tff(fact_5687_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_xw(set(A),fun(fun(B,A),fun(B,$o)),A3),F2))) ) ) ).

% finite_inverse_image
tff(fact_5688_distinct__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: 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,F2),Xb),Xs))
        <=> ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
            & distinct(A,Xs) ) ) ) ).

% distinct_insort
tff(fact_5689_finite__UNIV__surj__inj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),set(A),image2(A,A,F2),top_top(set(A))) = top_top(set(A)) )
       => inj_on(A,A,F2,top_top(set(A))) ) ) ).

% finite_UNIV_surj_inj
tff(fact_5690_finite__UNIV__inj__surj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( inj_on(A,A,F2,top_top(set(A)))
       => ( aa(set(A),set(A),image2(A,A,F2),top_top(set(A))) = top_top(set(A)) ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_5691_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3) ) ) ).

% inj_image_subset_iff
tff(fact_5692_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A11: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F7: fun(A,B)] :
            ( inj_on(A,B,F7,A3)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A3)),A11) )
      <=> ? [G5: fun(B,A)] : aa(set(B),set(A),image2(B,A,G5),A11) = A3 ) ) ).

% inj_on_iff_surj
tff(fact_5693_finite__surj__inj,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),image2(A,A,F2),A3))
       => inj_on(A,A,F2,A3) ) ) ).

% finite_surj_inj
tff(fact_5694_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3)
       => ( aa(set(B),$o,finite_finite2(B),B3)
         => aa(set(A),$o,finite_finite2(A),A3) ) ) ) ).

% inj_on_finite
tff(fact_5695_endo__inj__surj,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image2(A,A,F2),A3)),A3)
       => ( inj_on(A,A,F2,A3)
         => ( aa(set(A),set(A),image2(A,A,F2),A3) = A3 ) ) ) ) ).

% endo_inj_surj
tff(fact_5696_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C5: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
         => ( aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_5697_eq__card__imp__inj__on,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),nat,finite_card(A),A3) )
       => inj_on(A,B,F2,A3) ) ) ).

% eq_card_imp_inj_on
tff(fact_5698_inj__on__iff__eq__card,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( inj_on(A,B,F2,A3)
      <=> ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A3)) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_5699_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C5: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
         => ( aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,F2),B3)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_5700_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F2),A3))),aa(set(B),nat,finite_card(B),A3))
     => ~ inj_on(B,A,F2,A3) ) ).

% pigeonhole
tff(fact_5701_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: A,Xb: A,B2: A,F2: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A2,B2))
               => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),aa(A,B,F2,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,B2)) )
                  | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),aa(A,B,F2,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,A2)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_5702_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),Xb: B,B3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( member(B,Xb,aa(set(A),set(B),image2(A,B,F2),A3))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => member(A,the_inv_into(A,B,A3,F2,Xb),B3) ) ) ) ).

% the_inv_into_into
tff(fact_5703_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F2: fun(B,C)] :
      ( ! [I2: A,J2: A] :
          ( member(A,I2,I5)
         => ( member(A,J2,I5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,I2)),aa(A,set(B),A3,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I2)) ) ) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => inj_on(B,C,F2,aa(A,set(B),A3,I2)) )
       => inj_on(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_5704_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),F2: fun(B,C),A3: fun(A,set(B))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => inj_on(B,C,F2,aa(A,set(B),A3,I2)) )
       => inj_on(B,C,F2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5))) ) ) ).

% inj_on_INTER
tff(fact_5705_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),product_Pair(nat,nat,A2),B2))) ).

% le_prod_encode_2
tff(fact_5706_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),product_Pair(nat,nat,A2),B2))) ).

% le_prod_encode_1
tff(fact_5707_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S2: set(A),T3: set(B),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),T3)
       => ( ( aa(set(A),nat,finite_card(A),S2) = aa(set(B),nat,finite_card(B),T3) )
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),S2)),T3)
           => ( ! [X2: B] :
                  ( member(B,X2,T3)
                 => ? [Xa3: A] :
                      ( member(A,Xa3,S2)
                      & ( aa(A,B,F2,Xa3) = X2 ) ) )
            <=> inj_on(A,B,F2,S2) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_5708_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3)
       => ( inj_on(B,A,G,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),B3)),A3)
           => ( aa(set(A),$o,finite_finite2(A),A3)
             => ( aa(set(B),$o,finite_finite2(B),B3)
               => ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_5709_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),uminus_uminus(set(A)),A3))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F2),A3))) ) ).

% inj_image_Compl_subset
tff(fact_5710_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),C5: set(A),A3: set(C),B3: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F2,C5)
     => ( ! [X3: C] :
            ( member(C,X3,A3)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(C,set(A),B3,X3)),C5) )
       => ( member(C,J,A3)
         => ( aa(set(A),set(B),image2(A,B,F2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B3),A3))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_xx(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F2),B3)),A3)) ) ) ) ) ).

% image_INT
tff(fact_5711_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3))
         => ? [F3: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F3),A3)),B3)
              & inj_on(A,B,F3,A3) ) ) ) ) ).

% card_le_inj
tff(fact_5712_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3)
       => ( aa(set(B),$o,finite_finite2(B),B3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3)) ) ) ) ).

% card_inj_on_le
tff(fact_5713_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( ? [F7: fun(A,B)] :
              ( inj_on(A,B,F7,A3)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A3)),B3) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_5714_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_5715_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_xr(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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5716_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Mb: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K,Mb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_5717_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),$o)] :
      ( ? [T9: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S2))
          & aa(set(A),$o,P,T9) )
    <=> ? [T9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S2)
          & inj_on(B,A,F2,T9)
          & aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% ex_subset_image_inj
tff(fact_5718_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),$o)] :
      ( ! [T9: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S2))
         => aa(set(A),$o,P,T9) )
    <=> ! [T9: set(B)] :
          ( ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S2)
            & inj_on(B,A,F2,T9) )
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% all_subset_image_inj
tff(fact_5719_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_5720_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_5721_inj__Suc,axiom,
    ! [N3: set(nat)] : inj_on(nat,nat,suc,N3) ).

% inj_Suc
tff(fact_5722_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_5723_is__measure__trivial,axiom,
    ! [A: $tType,F2: fun(A,nat)] : fun_is_measure(A,F2) ).

% is_measure_trivial
tff(fact_5724_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_5725_inj__singleton,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,set(A),aTP_Lamp_mo(A,set(A)),A3) ).

% inj_singleton
tff(fact_5726_inj__on__diff__nat,axiom,
    ! [N3: set(nat),K: nat] :
      ( ! [N: nat] :
          ( member(nat,N,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) )
     => inj_on(nat,nat,aTP_Lamp_lf(nat,fun(nat,nat),K),N3) ) ).

% inj_on_diff_nat
tff(fact_5727_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,collect(set(nat),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_5728_measure__size,axiom,
    ! [A: $tType] :
      ( size(A)
     => fun_is_measure(A,size_size(A)) ) ).

% measure_size
tff(fact_5729_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_xz(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_5730_range__inj__infinite,axiom,
    ! [A: $tType,F2: fun(nat,A)] :
      ( inj_on(nat,A,F2,top_top(set(nat)))
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat)))) ) ).

% range_inj_infinite
tff(fact_5731_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [N: nat,F3: fun(nat,A)] :
          ( ( A3 = aa(set(nat),set(A),image2(nat,A,F3),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),N))) )
          & inj_on(nat,A,F3,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_5732_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [F3: fun(A,nat),N: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F3),A3) = collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),N)) )
          & inj_on(A,nat,F3,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_5733_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X3: nat] :
            ( member(nat,X3,I5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs)) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_5734_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_5735_infinite__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ? [F3: fun(nat,A)] :
          ( inj_on(nat,A,F3,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,F3),top_top(set(nat)))),S2) ) ) ).

% infinite_countable_subset
tff(fact_5736_inj__on__funpow__least,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A),S: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),S) = S )
     => ( ! [M2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M2),F2),S) != S ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_ya(fun(A,A),fun(A,fun(nat,A)),F2),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% inj_on_funpow_least
tff(fact_5737_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F2: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_yb(fun(B,list(A)),fun(B,set(A)),F2)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_5738_has__derivative__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(B,A),Xb: B,F9: fun(B,A),S2: set(B),Nb: int] :
          ( ( aa(B,A,F2,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F9,topolo174197925503356063within(B,Xb,S2))
           => has_derivative(B,A,aa(int,fun(B,A),aTP_Lamp_yc(fun(B,A),fun(int,fun(B,A)),F2),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_yd(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),F2),Xb),F9),Nb),topolo174197925503356063within(B,Xb,S2)) ) ) ) ).

% has_derivative_power_int
tff(fact_5739_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_ye(int,fun(A,A),Nb),aa(int,fun(A,A),aTP_Lamp_yf(A,fun(int,fun(A,A)),Xb),Nb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_power_int'
tff(fact_5740_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_5741_power__int__1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y: A] : power_int(A,Y,one_one(int)) = Y ) ).

% power_int_1_right
tff(fact_5742_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_5743_bind__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,list(A))] : bind(B,A,nil(B),F2) = nil(A) ).

% bind_simps(1)
tff(fact_5744_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W2: num,Y: A,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Y),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,Y,Mb)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_5745_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_5746_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_5747_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_5748_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_5749_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_5750_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_5751_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_5752_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_5753_power__int__minus1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y: A] : power_int(A,Y,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y) ) ).

% power_int_minus1_right
tff(fact_5754_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_5755_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_5756_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_5757_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_5758_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_5759_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_5760_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Mb: int] : power_int(A,divide_divide(A,Xb,Y),Mb) = divide_divide(A,power_int(A,Xb,Mb),power_int(A,Y,Mb)) ) ).

% power_int_divide_distrib
tff(fact_5761_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y),Mb) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),power_int(A,Y,Mb)) ) ).

% power_int_mult_distrib
tff(fact_5762_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_5763_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_5764_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_5765_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_5766_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_5767_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_5768_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),Nb: int] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S,aa(int,fun(A,B),aTP_Lamp_yg(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_on_power_int
tff(fact_5769_list__bind__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F2: fun(A,list(B)),G: fun(A,list(B))] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,list(B),F2,X3) = aa(A,list(B),G,X3) ) )
       => ( bind(A,B,Xs,F2) = bind(A,B,Ys,G) ) ) ) ).

% list_bind_cong
tff(fact_5770_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_5771_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_5772_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_5773_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_5774_tendsto__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),Nb: int] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aa(int,fun(A,B),aTP_Lamp_yh(fun(A,B),fun(int,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,power_int(B,A2,Nb)),F4) ) ) ) ).

% tendsto_power_int
tff(fact_5775_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,S: set(A),F2: fun(A,B),Nb: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),aa(int,fun(A,B),aTP_Lamp_yi(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_5776_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S: set(A),Nb: int] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,B,F2,Xb) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_yj(fun(A,B),fun(int,fun(A,B)),F2),Nb),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% differentiable_power_int
tff(fact_5777_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),Nb: int] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_th(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aa(int,fun(A,B),aTP_Lamp_yi(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_power_int
tff(fact_5778_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_5779_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(int,$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,Y,Nb)) ) ) ) ) ).

% power_int_mono
tff(fact_5780_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_5781_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_5782_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_5783_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_5784_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_5785_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_5786_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_5787_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_5788_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_5789_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_5790_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_5791_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_5792_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_5793_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_5794_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_5795_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S: set(A),Nb: int] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_yk(fun(A,A),fun(int,fun(A,A)),F2),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,F2,Xb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int))))),D2),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_power_int
tff(fact_5796_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_5797_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_5798_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ym(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_5799_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => member(B,aa(A,B,F2,X3),ring_1_Ints(B)) )
         => member(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3),ring_1_Ints(B)) ) ) ).

% Ints_sum
tff(fact_5800_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A3)
             => member(B,aa(A,B,F2,X3),ring_1_Ints(B)) )
         => member(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3),ring_1_Ints(B)) ) ) ).

% Ints_prod
tff(fact_5801_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_5802_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( ( member(A,Xb,ring_1_Ints(A))
            | member(A,Y,ring_1_Ints(A)) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_5803_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_5804_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_5805_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_5806_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_5807_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_5808_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_5809_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_5810_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_5811_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_5812_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_5813_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_5814_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_5815_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_5816_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A,P: fun(A,$o)] :
          ( member(A,Q2,ring_1_Ints(A))
         => ( ! [Z2: int] : aa(A,$o,P,aa(int,A,ring_1_of_int(A),Z2))
           => aa(A,$o,P,Q2) ) ) ) ).

% Ints_induct
tff(fact_5817_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_5818_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_5819_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : pow(Xb,one2) = Xb ).

% pow.simps(1)
tff(fact_5820_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_5821_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(set(A),$o,finite_finite2(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_yn(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_5822_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_5823_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( 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_5824_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_yo(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_5825_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_yp(A,$o)) ) ) ).

% Nats_altdef2
tff(fact_5826_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_5827_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_5828_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_5829_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_5830_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( Xb = Y )
            <=> aa(A,$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),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5831_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_5832_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_5833_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_5834_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_5835_pow_Osimps_I3_J,axiom,
    ! [Xb: num,Y: num] : pow(Xb,bit1(Y)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(Xb,Y))),Xb) ).

% pow.simps(3)
tff(fact_5836_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_xr(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_5837_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( 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) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X2: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X2,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X8)
                              & ! [X2: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X8) ) ),
                              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,$o))))),aTP_Lamp_yq(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5838_ball__empty,axiom,
    ! [A: $tType,P: fun(A,$o),X: A] :
      ( member(A,X,bot_bot(set(A)))
     => aa(A,$o,P,X) ) ).

% ball_empty
tff(fact_5839_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,P))
     => ( aa(set(B),$o,finite_finite2(B),collect(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_yr(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),P),Q)))
      <=> ! [Y4: A] :
            ( aa(A,$o,P,Y4)
           => aa(set(B),$o,finite_finite2(B),collect(B,aa(A,fun(B,$o),aTP_Lamp_ys(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),Y4))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_5840_ball__UNIV,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X2: A] :
          ( member(A,X2,top_top(set(A)))
         => aa(A,$o,P,X2) )
    <=> ! [X_13: A] : aa(A,$o,P,X_13) ) ).

% ball_UNIV
tff(fact_5841_sqr_Osimps_I2_J,axiom,
    ! [Nb: num] : sqr(bit0(Nb)) = bit0(bit0(sqr(Nb))) ).

% sqr.simps(2)
tff(fact_5842_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,$o)] : collect(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_yt(fun(B,A),fun(fun(B,$o),fun(A,$o)),F2),P)) = aa(set(B),set(A),image2(B,A,F2),collect(B,P)) ).

% setcompr_eq_image
tff(fact_5843_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : collect(A,aa(set(B),fun(A,$o),aTP_Lamp_yu(fun(B,A),fun(set(B),fun(A,$o)),F2),A3)) = aa(set(B),set(A),image2(B,A,F2),A3) ).

% Setcompr_eq_image
tff(fact_5844_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),F2: fun(A,fun(B,C))] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,P))
     => ( aa(set(B),$o,finite_finite2(B),collect(B,Q))
       => aa(set(C),$o,finite_finite2(C),collect(C,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_yv(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P),Q),F2))) ) ) ).

% finite_image_set2
tff(fact_5845_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,P))
     => aa(set(B),$o,finite_finite2(B),collect(B,aa(fun(A,B),fun(B,$o),aTP_Lamp_yw(fun(A,$o),fun(fun(A,B),fun(B,$o)),P),F2))) ) ).

% finite_image_set
tff(fact_5846_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_5847_finite_Omono,axiom,
    ! [A: $tType] : order_mono(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_yx(fun(set(A),$o),fun(set(A),$o))) ).

% finite.mono
tff(fact_5848_closed__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_yy(product_prod(A,A),$o))) ) ).

% closed_superdiagonal
tff(fact_5849_closed__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_yz(product_prod(A,A),$o))) ) ).

% closed_subdiagonal
tff(fact_5850_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_za(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_5851_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_zb(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_5852_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] : collect(A,aTP_Lamp_zc(fun(B,A),fun(A,$o),F2)) = aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_5853_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zd(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P),Net)
      <=> ! [X2: A] :
            ( member(A,X2,A3)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ys(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X2),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_5854_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [X3: A] :
            ( member(A,X3,A3)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ys(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X3),Net) )
       => eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zd(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P),Net) ) ) ).

% eventually_ball_finite
tff(fact_5855_sqr__conv__mult,axiom,
    ! [Xb: num] : sqr(Xb) = aa(num,num,aa(num,fun(num,num),times_times(num),Xb),Xb) ).

% sqr_conv_mult
tff(fact_5856_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A3) = aa(set(A),A,complete_Inf_Inf(A),collect(A,aTP_Lamp_ze(set(A),fun(A,$o),A3))) ) ).

% Sup_eq_Inf
tff(fact_5857_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A3) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aTP_Lamp_zf(set(A),fun(A,$o),A3))) ) ).

% Inf_eq_Sup
tff(fact_5858_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_zg(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_5859_pow_Osimps_I2_J,axiom,
    ! [Xb: num,Y: num] : pow(Xb,bit0(Y)) = sqr(pow(Xb,Y)) ).

% pow.simps(2)
tff(fact_5860_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A)] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S2)
           => ( aa(set(A),A,complete_Inf_Inf(A),S2) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aTP_Lamp_zh(set(A),fun(A,$o),S2))) ) ) ) ) ).

% cInf_cSup
tff(fact_5861_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A)] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S2)
           => ( aa(set(A),A,complete_Sup_Sup(A),S2) = aa(set(A),A,complete_Inf_Inf(A),collect(A,aTP_Lamp_zi(set(A),fun(A,$o),S2))) ) ) ) ) ).

% cSup_cInf
tff(fact_5862_sqr_Osimps_I3_J,axiom,
    ! [Nb: num] : sqr(bit1(Nb)) = bit1(bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(Nb)),Nb))) ).

% sqr.simps(3)
tff(fact_5863_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Deg2: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya),Deg2)
    <=> ( ( Dega = Deg2 )
        & $let(
            n: nat,
            n:= divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),bit0(one2))),
            $let(
              m2: nat,
              m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Dega),n),
              ( ! [X2: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                 => vEBT_VEBT_valid(X2,n) )
              & vEBT_VEBT_valid(Summarya,m2)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X8)
                  & ! [X2: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X8) ) ),
                  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,$o))))),aTP_Lamp_yq(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Dega),TreeLista),Summarya),n),m2)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5864_funpow__inj__finite,axiom,
    ! [A: $tType,P3: fun(A,A),Xb: A] :
      ( inj_on(A,A,P3,top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_zj(fun(A,A),fun(A,fun(A,$o)),P3),Xb)))
       => ~ ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),P3),Xb) != Xb ) ) ) ) ).

% funpow_inj_finite
tff(fact_5865_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))),
                    $let(
                      m2: nat,
                      m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                      ( ! [X2: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_VEBT_valid(X2,n) )
                      & vEBT_VEBT_valid(Summary,m2)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X8)
                          & ! [X2: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X8) ) ),
                          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,$o))))),aTP_Lamp_yq(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5866_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))),
                      $let(
                        m2: nat,
                        m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                        ( ! [X2: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_VEBT_valid(X2,n) )
                        & vEBT_VEBT_valid(Summary,m2)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X8)
                            & ! [X2: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X8) ) ),
                            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,$o))))),aTP_Lamp_yq(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_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),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),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),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))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X2: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X2,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X8)
                              & ! [X2: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X8) ) ),
                              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,$o))))),aTP_Lamp_yq(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$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),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),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),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))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X2: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X2,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X8)
                                & ! [X2: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X8) ) ),
                                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,$o))))),aTP_Lamp_yq(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$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,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X2: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X2,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X8)
                                & ! [X2: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X8) ) ),
                                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,$o))))),aTP_Lamp_yq(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg,TreeList,Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_5870_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_zk(set(set(A)),fun(set(A),$o),A3))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A3))) ) ).

% Sup_Inf_le
tff(fact_5871_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_zl(set(set(A)),fun(set(A),$o),A3))))) ) ).

% Inf_Sup_le
tff(fact_5872_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A3: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_zm(set(set(A)),fun(set(A),$o),A3))))) ) ).

% finite_Inf_Sup
tff(fact_5873_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q: fun(A,fun(B,C)),F2: fun(D,B)] :
          ( order_mono(A,fun(B,C),Q)
         => order_mono(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_zn(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F2)) ) ) ).

% mono_compose
tff(fact_5874_Pow__Compl,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A3)) = collect(set(A),aTP_Lamp_zo(set(A),fun(set(A),$o),A3)) ).

% Pow_Compl
tff(fact_5875_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)),collect(set(A),aTP_Lamp_zp(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_5876_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_zq(A,$o)) ) ) ).

% Nats_altdef1
tff(fact_5877_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A)] : order_mono(fun(A,$o),fun(A,$o),aTP_Lamp_zr(fun(A,A),fun(fun(A,$o),fun(A,$o)),F2)) ) ).

% iteratesp.mono
tff(fact_5878_Ball__Collect,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( ! [X2: A] :
          ( member(A,X2,A3)
         => aa(A,$o,P,X2) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,P)) ) ).

% Ball_Collect
tff(fact_5879_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_5880_UnCI,axiom,
    ! [A: $tType,C2: A,B3: set(A),A3: set(A)] :
      ( ( ~ member(A,C2,B3)
       => member(A,C2,A3) )
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ).

% UnCI
tff(fact_5881_Un__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
    <=> ( member(A,C2,A3)
        | member(A,C2,B3) ) ) ).

% Un_iff
tff(fact_5882_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_5883_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% sup.bounded_iff
tff(fact_5884_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_5885_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_5886_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) )
        <=> ( ( Xb = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_5887_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = bot_bot(A) )
        <=> ( ( Xb = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_5888_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_5889_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_5890_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_5891_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_5892_Un__empty,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = bot_bot(set(A)) )
    <=> ( ( A3 = bot_bot(set(A)) )
        & ( B3 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_5893_finite__Un,axiom,
    ! [A: $tType,F4: set(A),G3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G3))
    <=> ( aa(set(A),$o,finite_finite2(A),F4)
        & aa(set(A),$o,finite_finite2(A),G3) ) ) ).

% finite_Un
tff(fact_5894_Un__subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5) ) ) ).

% Un_subset_iff
tff(fact_5895_Un__insert__left,axiom,
    ! [A: $tType,A2: A,B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)),C5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)) ).

% Un_insert_left
tff(fact_5896_Un__insert__right,axiom,
    ! [A: $tType,A3: set(A),A2: A,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ).

% Un_insert_right
tff(fact_5897_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_5898_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)) = S2 ).

% Int_Un_eq(3)
tff(fact_5899_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_5900_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)),S2) = S2 ).

% Int_Un_eq(1)
tff(fact_5901_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_5902_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)) = S2 ).

% Un_Int_eq(3)
tff(fact_5903_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_5904_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)),S2) = S2 ).

% Un_Int_eq(1)
tff(fact_5905_Un__Diff__cancel2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3) ).

% Un_Diff_cancel2
tff(fact_5906_Un__Diff__cancel,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) ).

% Un_Diff_cancel
tff(fact_5907_Compl__Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),B3) ).

% Compl_Diff_eq
tff(fact_5908_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F2: fun(B,A),G: fun(B,A),S2: set(B)] : aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_zs(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G)),S2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),collect(B,P)))),aa(set(B),set(A),image2(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),collect(B,aTP_Lamp_zt(fun(B,$o),fun(B,$o),P))))) ).

% if_image_distrib
tff(fact_5909_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),union(A),Xs),Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_union
tff(fact_5910_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B3: 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_zu(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),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),A3),C5))),B3)) ).

% UN_simps(2)
tff(fact_5911_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: 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_zv(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),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)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C5)))) ).

% UN_simps(3)
tff(fact_5912_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),A3: A,B3: A] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A3)),aa(A,B,F2,B3))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3))) ) ) ).

% mono_sup
tff(fact_5913_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_5914_Diff__partition,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_5915_Diff__subset__conv,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),C5)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)) ) ).

% Diff_subset_conv
tff(fact_5916_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_5917_singleton__Un__iff,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_5918_Un__singleton__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),Xb: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) )
    <=> ( ( ( A3 = bot_bot(set(A)) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) )
          & ( B3 = bot_bot(set(A)) ) )
        | ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) )
          & ( B3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_5919_insert__is__Un,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))),A3) ).

% insert_is_Un
tff(fact_5920_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_5921_cSup__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A3)
           => ( ( B3 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,B3)
               => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_5922_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_5923_Un__empty__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B3) = B3 ).

% Un_empty_left
tff(fact_5924_Un__empty__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),bot_bot(set(A))) = A3 ).

% Un_empty_right
tff(fact_5925_chain__empty,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).

% chain_empty
tff(fact_5926_Un__UNIV__right,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_5927_Un__UNIV__left,axiom,
    ! [A: $tType,B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B3) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_5928_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_5929_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_5930_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_5931_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_5932_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% sup.strict_boundedE
tff(fact_5933_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_5934_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_5935_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_5936_insert__def,axiom,
    ! [A: $tType,A2: A,B3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,aTP_Lamp_bm(A,fun(A,$o),A2))),B3) ).

% insert_def
tff(fact_5937_UnE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
     => ( ~ member(A,C2,A3)
       => member(A,C2,B3) ) ) ).

% UnE
tff(fact_5938_UnI1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B3: set(A)] :
      ( member(A,C2,A3)
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ).

% UnI1
tff(fact_5939_UnI2,axiom,
    ! [A: $tType,C2: A,B3: set(A),A3: set(A)] :
      ( member(A,C2,B3)
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ).

% UnI2
tff(fact_5940_bex__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,$o)] :
      ( ? [X2: A] :
          ( member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
          & aa(A,$o,P,X2) )
    <=> ( ? [X2: A] :
            ( member(A,X2,A3)
            & aa(A,$o,P,X2) )
        | ? [X2: A] :
            ( member(A,X2,B3)
            & aa(A,$o,P,X2) ) ) ) ).

% bex_Un
tff(fact_5941_ball__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,$o)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
         => aa(A,$o,P,X2) )
    <=> ( ! [X2: A] :
            ( member(A,X2,A3)
           => aa(A,$o,P,X2) )
        & ! [X2: A] :
            ( member(A,X2,B3)
           => aa(A,$o,P,X2) ) ) ) ).

% ball_Un
tff(fact_5942_Un__assoc,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)) ).

% Un_assoc
tff(fact_5943_Un__absorb,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),A3) = A3 ).

% Un_absorb
tff(fact_5944_Un__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3) ).

% Un_commute
tff(fact_5945_Un__left__absorb,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) ).

% Un_left_absorb
tff(fact_5946_Un__left__commute,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C5)) ).

% Un_left_commute
tff(fact_5947_Un__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),C5)) ).

% Un_Diff
tff(fact_5948_Un__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_zw(set(A),fun(set(A),fun(A,$o)),A3),B3)) ).

% Un_def
tff(fact_5949_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,P)),collect(A,Q)) ).

% Collect_disj_eq
tff(fact_5950_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_zx(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P))),collect(A,Q)) ).

% Collect_imp_eq
tff(fact_5951_Un__Int__crazy,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C5),A3)) ).

% Un_Int_crazy
tff(fact_5952_Int__Un__distrib,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5)) ).

% Int_Un_distrib
tff(fact_5953_Un__Int__distrib,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C5)) ).

% Un_Int_distrib
tff(fact_5954_Int__Un__distrib2,axiom,
    ! [A: $tType,B3: set(A),C5: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),A3)) ).

% Int_Un_distrib2
tff(fact_5955_Un__Int__distrib2,axiom,
    ! [A: $tType,B3: set(A),C5: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C5),A3)) ).

% Un_Int_distrib2
tff(fact_5956_infinite__Un,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3))
    <=> ( ~ aa(set(A),$o,finite_finite2(A),S2)
        | ~ aa(set(A),$o,finite_finite2(A),T3) ) ) ).

% infinite_Un
tff(fact_5957_Un__infinite,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( ~ aa(set(A),$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),T3)) ) ).

% Un_infinite
tff(fact_5958_finite__UnI,axiom,
    ! [A: $tType,F4: set(A),G3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,finite_finite2(A),G3)
       => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G3)) ) ) ).

% finite_UnI
tff(fact_5959_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,$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),Y)),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),Y),Z))) ) ).

% distrib_inf_le
tff(fact_5960_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,$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),Y),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z))) ) ).

% distrib_sup_le
tff(fact_5961_Un__Pow__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A3)),pow2(A,B3))),pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ).

% Un_Pow_subset
tff(fact_5962_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% inf_sup_ord(4)
tff(fact_5963_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% inf_sup_ord(3)
tff(fact_5964_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_5965_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_5966_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% sup_ge1
tff(fact_5967_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% sup_ge2
tff(fact_5968_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_5969_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_5970_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_5971_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$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),C2),D2)) ) ) ) ).

% sup_mono
tff(fact_5972_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,$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),Y),Z)),Xb) ) ) ) ).

% sup_least
tff(fact_5973_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_5974_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_5975_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_5976_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X3: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F2,X3),Y3))
         => ( ! [X3: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F2,X3),Y3))
           => ( ! [X3: A,Y3: A,Z2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y3),Z2)),X3) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F2,Xb),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_5977_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_5978_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_5979_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Xb ) ) ) ).

% sup_absorb1
tff(fact_5980_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_5981_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% sup.boundedE
tff(fact_5982_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2) ) ) ) ).

% sup.boundedI
tff(fact_5983_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_5984_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_5985_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_5986_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_5987_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_5988_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI1
tff(fact_5989_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI2
tff(fact_5990_Un__mono,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),B3: set(A),D5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),D5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C5),D5)) ) ) ).

% Un_mono
tff(fact_5991_Un__least,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C5) ) ) ).

% Un_least
tff(fact_5992_Un__upper1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ).

% Un_upper1
tff(fact_5993_Un__upper2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ).

% Un_upper2
tff(fact_5994_Un__absorb1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_5995_Un__absorb2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_5996_subset__UnE,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
     => ~ ! [A10: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),A3)
           => ! [B13: set(A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B13),B3)
               => ( C5 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A10),B13) ) ) ) ) ).

% subset_UnE
tff(fact_5997_subset__Un__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_5998_chain__subset,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o)),A3: set(A),B3: set(A)] :
      ( comple1602240252501008431_chain(A,Ord,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => comple1602240252501008431_chain(A,Ord,B3) ) ) ).

% chain_subset
tff(fact_5999_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),A3) ) ).

% Un_Int_assoc_eq
tff(fact_6000_Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C5)) ).

% Diff_Un
tff(fact_6001_Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C5)) ).

% Diff_Int
tff(fact_6002_Int__Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)) = A3 ).

% Int_Diff_Un
tff(fact_6003_Un__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = A3 ).

% Un_Diff_Int
tff(fact_6004_Compl__partition2,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),A3) = top_top(set(A)) ).

% Compl_partition2
tff(fact_6005_Compl__partition,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3)) = top_top(set(A)) ).

% Compl_partition
tff(fact_6006_cSUP__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( ( B3 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),B3))
               => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),B3))) ) ) ) ) ) ) ).

% cSUP_union
tff(fact_6007_image__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B3: set(B)] : aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),aa(set(B),set(A),image2(B,A,F2),B3)) ).

% image_Un
tff(fact_6008_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A),Z: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(A,$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),A3)),Z) ) ) ) ).

% ccpo_Sup_least
tff(fact_6009_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A),Xb: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( member(A,Xb,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).

% ccpo_Sup_upper
tff(fact_6010_Compl__Un,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Compl_Un
tff(fact_6011_Compl__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B3)) ).

% Compl_Int
tff(fact_6012_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_6013_mono__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A3: set(A),B3: set(A)] :
      ( order_mono(set(A),set(B),F2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(A),set(B),F2,A3)),aa(set(A),set(B),F2,B3))),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ) ).

% mono_Un
tff(fact_6014_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% sup_shunt
tff(fact_6015_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q2),R2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q2))),R2) ) ) ).

% sup_neg_inf
tff(fact_6016_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,uminus_uminus(A),Y))),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ) ).

% shunt2
tff(fact_6017_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Z)
        <=> aa(A,$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),Y)),Z)) ) ) ).

% shunt1
tff(fact_6018_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A2: A,Xb: A,Y: 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),Y) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Y) = top_top(A) )
               => ( Xb = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_6019_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] :
                  ( member(A,X3,A3)
                 => ( member(A,Y3,A3)
                   => member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3),A3) ) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),A3),A3) ) ) ) ) ).

% finite_Sup_in
tff(fact_6020_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] : aa(A,$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),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ).

% less_eq_Inf_inter
tff(fact_6021_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_6022_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_6023_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),bot_bot(set(B))))) = A3 ).

% Union_image_empty
tff(fact_6024_card__Un__le,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(nat,$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)),A3),B3))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))) ).

% card_Un_le
tff(fact_6025_atLeastLessThan__add__Un,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( set_or7035219750837199246ssThan(nat,Ia,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,Ia,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_6026_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_6027_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_6028_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_6029_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_6030_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)),insert2(A),Xb),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_6031_Pow__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : pow2(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A3)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2)),pow2(A,A3))) ).

% Pow_insert
tff(fact_6032_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B3: 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),A3),C5))),B3) = $ite(C5 = bot_bot(set(B)),B3,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_zu(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B3)),C5))) ).

% UN_extend_simps(2)
tff(fact_6033_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B3),C5))) = $ite(C5 = bot_bot(set(B)),A3,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_zv(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B3)),C5))) ).

% UN_extend_simps(3)
tff(fact_6034_Inter__Un__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B3))) ).

% Inter_Un_subset
tff(fact_6035_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),Xb) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_6036_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)),insert2(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_6037_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)),insert2(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_6038_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_6039_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ).

% sum.union_inter
tff(fact_6040_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ).

% prod.union_inter
tff(fact_6041_card__Un__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% card_Un_Int
tff(fact_6042_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_6043_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_6044_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_6045_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_6046_cInf__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A3)
           => ( ( B3 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,B3)
               => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_6047_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_6048_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B3)) ) ) ) ) ) ) ).

% Max.union
tff(fact_6049_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_6050_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_6051_ivl__disj__un__singleton_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = aa(A,set(A),set_ord_atLeast(A),L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_6052_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_6053_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_mm(A,fun(nat,A),B3)),collect(nat,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ).

% SUP_nat_binary
tff(fact_6054_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_6055_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),A3))
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),A3))) = 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_zy(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A3)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_6056_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),G: fun(A,B),B3: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( inj_on(A,B,G,B3)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),aa(set(A),set(B),image2(A,B,G),B3)) = bot_bot(set(B)) )
         => inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_zz(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_6057_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_aaa(A,fun(A,$o)),aTP_Lamp_aab(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_6058_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( condit941137186595557371_above(A,A3)
         => ( condit941137186595557371_above(A,B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) != bot_bot(set(A)) )
             => aa(A,$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)),A3),B3))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_6059_cSUP__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),A3))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A2)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ).

% cSUP_insert
tff(fact_6060_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
                 => ( aa(A,B,G,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_6061_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ).

% sum_Un
tff(fact_6062_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B3)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6063_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
                 => ( aa(A,B,G,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6064_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B3)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_6065_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_6066_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% sum_Un2
tff(fact_6067_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ).

% sum.union_diff2
tff(fact_6068_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ).

% prod.union_diff2
tff(fact_6069_card__Un__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_6070_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
    <=> ( inj_on(A,B,F2,A3)
        & inj_on(A,B,F2,B3)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B3))),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_6071_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_6072_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_6073_cINF__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),B3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A3))
           => ( ( B3 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),B3))
               => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),B3))) ) ) ) ) ) ) ).

% cINF_union
tff(fact_6074_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_6075_sum__Un__nat,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B3))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_6076_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_6077_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_6078_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),B3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
                 => ( aa(A,B,F2,X3) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = divide_divide(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ) ) ).

% prod_Un
tff(fact_6079_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),A3),A3) ) ) ) ) ).

% in_chain_finite
tff(fact_6080_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C5: set(A),B3: set(A),Xb: A] :
      ( inj_on(A,B,G,C5)
     => ( aa(set(A),$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)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(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_aac(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),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)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_6081_Pow__fold,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( pow2(A,A3) = finite_fold(A,set(set(A)),aTP_Lamp_aad(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A3) ) ) ).

% Pow_fold
tff(fact_6082_fold__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Z: A] : finite_fold(B,A,F2,Z,bot_bot(set(B))) = Z ).

% fold_empty
tff(fact_6083_fold__infinite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( finite_fold(A,B,F2,Z,A3) = Z ) ) ).

% fold_infinite
tff(fact_6084_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_6085_sup__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = collect(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A3)),aTP_Lamp_a(set(A),fun(A,$o),B3))) ).

% sup_set_def
tff(fact_6086_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_6087_fold__closed__eq,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),Z: B] :
      ( ! [A4: A,B4: B] :
          ( member(A,A4,A3)
         => ( member(B,B4,B3)
           => ( aa(B,B,aa(A,fun(B,B),F2,A4),B4) = aa(B,B,aa(A,fun(B,B),G,A4),B4) ) ) )
     => ( ! [A4: A,B4: B] :
            ( member(A,A4,A3)
           => ( member(B,B4,B3)
             => member(B,aa(B,B,aa(A,fun(B,B),G,A4),B4),B3) ) )
       => ( member(B,Z,B3)
         => ( finite_fold(A,B,F2,Z,A3) = finite_fold(A,B,G,Z,A3) ) ) ) ) ).

% fold_closed_eq
tff(fact_6088_union__fold__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = finite_fold(A,set(A),insert2(A),B3,A3) ) ) ).

% union_fold_insert
tff(fact_6089_sup__Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B3) = finite_fold(A,A,sup_sup(A),B3,A3) ) ) ) ).

% sup_Sup_fold_sup
tff(fact_6090_inf__Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A3)),B3) = finite_fold(A,A,inf_inf(A),B3,A3) ) ) ) ).

% inf_Inf_fold_inf
tff(fact_6091_fold__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A3: set(A),F2: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G,A3)
     => ( finite_fold(B,C,F2,Z,aa(set(A),set(B),image2(A,B,G),A3)) = finite_fold(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F2),G),Z,A3) ) ) ).

% fold_image
tff(fact_6092_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,complete_Sup_Sup(A),A3) = finite_fold(A,A,sup_sup(A),bot_bot(A),A3) ) ) ) ).

% Sup_fold_sup
tff(fact_6093_Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,complete_Inf_Inf(A),A3) = finite_fold(A,A,inf_inf(A),top_top(A),A3) ) ) ) ).

% Inf_fold_inf
tff(fact_6094_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A3) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G),zero_zero(A),A3) ) ).

% sum.eq_fold
tff(fact_6095_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = finite_fold(A,A,ord_max(A),Xb,A3) ) ) ) ).

% Max.eq_fold
tff(fact_6096_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),set(B),image2(A,B,F2),A3) = finite_fold(A,set(B),aTP_Lamp_aae(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),A3) ) ) ).

% image_fold_insert
tff(fact_6097_sup__SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),B3: B,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),B3),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F2),B3,A3) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_6098_inf__INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),B3: B,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),B3),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F2),B3,A3) ) ) ) ).

% inf_INF_fold_inf
tff(fact_6099_SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F2),bot_bot(B),A3) ) ) ) ).

% SUP_fold_sup
tff(fact_6100_INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F2),top_top(B),A3) ) ) ) ).

% INF_fold_inf
tff(fact_6101_flat__lub__def,axiom,
    ! [A: $tType,B2: A,A3: set(A)] :
      partial_flat_lub(A,B2,A3) = $ite(aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),B2,the(A,aa(set(A),fun(A,$o),aTP_Lamp_aaf(A,fun(set(A),fun(A,$o)),B2),A3))) ).

% flat_lub_def
tff(fact_6102_Set__filter__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( filter3(A,P,A3) = finite_fold(A,set(A),aTP_Lamp_aag(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A3) ) ) ).

% Set_filter_fold
tff(fact_6103_member__filter,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),A3: set(A)] :
      ( member(A,Xb,filter3(A,P,A3))
    <=> ( member(A,Xb,A3)
        & aa(A,$o,P,Xb) ) ) ).

% member_filter
tff(fact_6104_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A3: set(A)] : filter3(A,P,A3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_aah(fun(A,$o),fun(set(A),fun(A,$o)),P),A3)) ).

% Set.filter_def
tff(fact_6105_finite__filter,axiom,
    ! [A: $tType,S2: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => aa(set(A),$o,finite_finite2(A),filter3(A,P,S2)) ) ).

% finite_filter
tff(fact_6106_Ball__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [X2: A] :
            ( member(A,X2,A3)
           => aa(A,$o,P,X2) )
      <=> finite_fold(A,$o,aTP_Lamp_aai(fun(A,$o),fun(A,fun($o,$o)),P),$true,A3) ) ) ).

% Ball_fold
tff(fact_6107_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B3: set(A),Xb: B,A3: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),image2(A,set(product_prod(B,A)),aTP_Lamp_aaj(B,fun(A,set(product_prod(B,A))),Xb)),B3))),A3) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_aak(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xb),A3,B3) ) ) ).

% fold_union_pair
tff(fact_6108_card_Oeq__fold,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = finite_fold(A,nat,aTP_Lamp_aal(A,fun(nat,nat)),zero_zero(nat),A3) ).

% card.eq_fold
tff(fact_6109_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_xr(A,A)),nil(A),A3) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_6110_inter__Set__filter,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = filter3(A,aTP_Lamp_a(set(A),fun(A,$o),A3),B3) ) ) ).

% inter_Set_filter
tff(fact_6111_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_aam(real,$o)) ).

% Rats_eq_int_div_nat
tff(fact_6112_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_6113_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_6114_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_6115_Rats__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => member(A,zero_zero(A),field_char_0_Rats(A)) ) ).

% Rats_0
tff(fact_6116_Rats__dense__in__real,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => ? [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),Y) ) ) ).

% Rats_dense_in_real
tff(fact_6117_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_6118_Ints__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),ring_1_Ints(A)),field_char_0_Rats(A)) ) ).

% Ints_subset_Rats
tff(fact_6119_Nats__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),semiring_1_Nats(A)),field_char_0_Rats(A)) ) ).

% Nats_subset_Rats
tff(fact_6120_Func__map,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,G: fun(A,B),A25: set(A),A14: set(B),F1: fun(B,C),B1: set(C),F22: fun(D,A),B22: set(D)] :
      ( member(fun(A,B),G,bNF_Wellorder_Func(A,B,A25,A14))
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F1),A14)),B1)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(D),set(A),image2(D,A,F22),B22)),A25)
         => member(fun(D,C),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B22,F1,F22),G),bNF_Wellorder_Func(D,C,B22,B1)) ) ) ) ).

% Func_map
tff(fact_6121_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,A3,B3) = bot_bot(set(fun(A,B))) )
    <=> ( ( A3 != bot_bot(set(A)) )
        & ( B3 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_6122_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A3: set(B)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,A3,B3) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_6123_Id__on__fold,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( id_on(A,A3) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_aan(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A3) ) ) ).

% Id_on_fold
tff(fact_6124_Id__on__def,axiom,
    ! [A: $tType,A3: set(A)] : id_on(A,A3) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),image2(A,set(product_prod(A,A)),aTP_Lamp_aao(A,set(product_prod(A,A)))),A3)) ).

% Id_on_def
tff(fact_6125_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_6126_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B3)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_aap(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B3)) ) ).

% comp_fun_commute_product_fold
tff(fact_6127_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),Ia: B,B3: set(A),J4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),fun_upd(B,set(A),A3,Ia,B3)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Ia),bot_bot(set(B))))))),
        $ite(member(B,Ia,J4),B3,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_6128_comp__fun__commute_Ointro,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( ! [Y3: A,X3: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y3))
     => finite6289374366891150609ommute(A,B,F2) ) ).

% comp_fun_commute.intro
tff(fact_6129_comp__fun__commute_Ocomp__fun__commute,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),Y: A,Xb: A] :
      ( finite6289374366891150609ommute(A,B,F2)
     => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),aa(A,fun(B,B),F2,Xb)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xb)),aa(A,fun(B,B),F2,Y)) ) ) ).

% comp_fun_commute.comp_fun_commute
tff(fact_6130_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(A,fun(B,B)),G: fun(C,A)] :
      ( finite6289374366891150609ommute(A,B,F2)
     => finite6289374366891150609ommute(C,B,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F2),G)) ) ).

% comp_fun_commute.comp_comp_fun_commute
tff(fact_6131_comp__fun__commute__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F2)
    <=> ! [Y4: A,X2: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),aa(A,fun(B,B),F2,X2)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X2)),aa(A,fun(B,B),F2,Y4)) ) ).

% comp_fun_commute_def
tff(fact_6132_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_aaq(fun(B,B),fun(A,fun(B,B)),F2)) ).

% comp_fun_commute_const
tff(fact_6133_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C2: B,P: fun(fun(A,B),$o)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aa(B,fun(A,$o),aTP_Lamp_aar(fun(A,B),fun(B,fun(A,$o)),F2),C2)))
     => ( aa(fun(A,B),$o,P,aTP_Lamp_lv(B,fun(A,B),C2))
       => ( ! [A4: A,B4: B,F3: fun(A,B)] :
              ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aas(B,fun(fun(A,B),fun(A,$o)),C2),F3)))
             => ( ( aa(A,B,F3,A4) = C2 )
               => ( ( B4 != C2 )
                 => ( aa(fun(A,B),$o,P,F3)
                   => aa(fun(A,B),$o,P,fun_upd(A,B,F3,A4,B4)) ) ) ) )
         => aa(fun(A,B),$o,P,F2) ) ) ) ).

% finite_update_induct
tff(fact_6134_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P: fun(A,$o)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_aag(fun(A,$o),fun(A,fun(set(A),set(A))),P)) ).

% comp_fun_commute_filter_fold
tff(fact_6135_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F2)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_aat(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F2),G)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_6136_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xb: B,Y: A,A3: set(B)] :
      aa(set(B),set(A),image2(B,A,fun_upd(B,A,F2,Xb,Y)),A3) = $ite(member(B,Xb,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xb),bot_bot(set(B)))))),aa(set(B),set(A),image2(B,A,F2),A3)) ).

% fun_upd_image
tff(fact_6137_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)),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aTP_Lamp_aav(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_6138_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aaw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_aax(set(product_prod(A,A)),fun(nat,$o),R))))) ) ) ).

% relpow_finite_bounded1
tff(fact_6139_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),B3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),S2)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = finite_fold(A,B,F2,finite_fold(A,B,F2,Z,A3),B3) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_6140_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_6141_relpow__0__E,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(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))),zero_zero(nat)),R))
     => ( Xb = Y ) ) ).

% relpow_0_E
tff(fact_6142_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),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_6143_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => finite4664212375090638736ute_on(A,B,S2,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_aat(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F2),G)) ) ).

% comp_fun_commute_on.comp_fun_commute_on_funpow
tff(fact_6144_comp__fun__commute__on_Ofun__left__comm,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,Y: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( member(A,Xb,S2)
       => ( member(A,Y,S2)
         => ( aa(B,B,aa(A,fun(B,B),F2,Y),aa(B,B,aa(A,fun(B,B),F2,Xb),Z)) = aa(B,B,aa(A,fun(B,B),F2,Xb),aa(B,B,aa(A,fun(B,B),F2,Y),Z)) ) ) ) ) ).

% comp_fun_commute_on.fun_left_comm
tff(fact_6145_comp__fun__commute__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
    <=> ! [X2: A,Y4: A] :
          ( member(A,X2,S2)
         => ( member(A,Y4,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),aa(A,fun(B,B),F2,X2)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X2)),aa(A,fun(B,B),F2,Y4)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_6146_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,Y: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( member(A,Xb,S2)
       => ( member(A,Y,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),aa(A,fun(B,B),F2,Xb)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xb)),aa(A,fun(B,B),F2,Y)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_6147_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,Y: A,G: fun(C,B)] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( member(A,Xb,S2)
       => ( member(A,Y,S2)
         => ( aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Y)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Xb)),G)) = aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Xb)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Y)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_6148_comp__fun__commute__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,S2)
         => ( member(A,Y3,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y3)) ) ) )
     => finite4664212375090638736ute_on(A,B,S2,F2) ) ).

% comp_fun_commute_on.intro
tff(fact_6149_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_xr(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_6150_comp__fun__commute__def_H,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F2)
    <=> finite4664212375090638736ute_on(A,B,top_top(set(A)),F2) ) ).

% comp_fun_commute_def'
tff(fact_6151_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),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 ) )
       => ~ ! [Y3: A,M2: nat] :
              ( ( Nb = aa(nat,nat,suc,M2) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y3),R)
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M2),R)) ) ) ) ) ).

% relpow_E2
tff(fact_6152_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),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 ) )
       => ~ ! [Y3: A,M2: nat] :
              ( ( Nb = aa(nat,nat,suc,M2) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),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))),M2),R))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z),R) ) ) ) ) ).

% relpow_E
tff(fact_6153_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_6154_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),A3: set(A),S: B,Ta: B,B3: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( finite4664212375090638736ute_on(A,B,S2,G)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S2)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( ! [X3: A] :
                  ( member(A,X3,A3)
                 => ( aa(A,fun(B,B),F2,X3) = aa(A,fun(B,B),G,X3) ) )
             => ( ( S = Ta )
               => ( ( A3 = B3 )
                 => ( finite_fold(A,B,F2,S,A3) = finite_fold(A,B,G,Ta,B3) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_6155_fold__atLeastAtMost__nat,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      ( finite6289374366891150609ommute(nat,A,F2)
     => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc2) = finite_fold(nat,A,F2,Acc2,set_or1337092689740270186AtMost(nat,A2,B2)) ) ) ).

% fold_atLeastAtMost_nat
tff(fact_6156_relpow__fun__conv,axiom,
    ! [A: $tType,A2: A,B2: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = A2 )
          & ( aa(nat,A,F7,Nb) = B2 )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
             => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F7,I4)),aa(nat,A,F7,aa(nat,nat,suc,I4))),R) ) ) ) ).

% relpow_fun_conv
tff(fact_6157_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F2: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G),top_top(set(C)))),S2)
       => finite4664212375090638736ute_on(C,B,R,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F2),G)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_6158_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(B,B,aa(A,fun(B,B),F2,Xb),finite_fold(A,B,F2,Z,A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xb),Z),A3) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_6159_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xb),Z),A3) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_6160_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(A,fun(B,B),F2,Xb),finite_fold(A,B,F2,Z,A3)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_6161_relpow__finite__bounded,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aaw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_aay(set(product_prod(A,A)),fun(nat,$o),R))))) ) ).

% relpow_finite_bounded
tff(fact_6162_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_aad(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_6163_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Xb: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( finite_fold(A,B,F2,Z,A3) = aa(B,B,aa(A,fun(B,B),F2,Xb),finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_6164_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(A,fun(B,B),F2,Xb),finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_6165_ntrancl__def,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,Nb,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_aaw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_aaz(nat,fun(nat,$o),Nb)))) ).

% ntrancl_def
tff(fact_6166_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),A2: B,B2: A] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F2),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),F2,A2,aa(A,option(A),some(A),B2))),top_top(set(B)))) ) ).

% finite_range_updI
tff(fact_6167_ntrancl__Zero,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R) = R ).

% ntrancl_Zero
tff(fact_6168_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_aaw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_aax(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_6169_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Xb: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),Mb,Xb,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),insert2(A),Xb),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_6170_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_6171_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_6172_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_6173_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,Xb: A,D5: set(A),Mb: fun(A,option(B)),Y: option(B)] :
      ( member(A,Xb,D5)
     => ( fun_upd(A,option(B),restrict_map(A,B,Mb,D5),Xb,Y) = 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)),insert2(A),Xb),bot_bot(set(A))))),Xb,Y) ) ) ).

% fun_upd_restrict_conv
tff(fact_6174_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Xb: A,Y: option(B),D5: set(A)] :
      restrict_map(A,B,fun_upd(A,option(B),Mb,Xb,Y),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)),insert2(A),Xb),bot_bot(set(A))))),Xb,Y),restrict_map(A,B,Mb,D5)) ).

% restrict_fun_upd
tff(fact_6175_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)),insert2(A),Xb),bot_bot(set(A))))),restrict_map(A,B,Mb,D5)) ).

% fun_upd_None_restrict
tff(fact_6176_trancl__power,axiom,
    ! [A: $tType,P3: product_prod(A,A),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P3,transitive_trancl(A,R))
    <=> ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
          & member(product_prod(A,A),P3,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N2),R)) ) ) ).

% trancl_power
tff(fact_6177_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_6178_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),D5: set(A),Xb: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,Mb,D5),Xb,Y) = 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)),insert2(A),Xb),bot_bot(set(A))))),Xb,Y) ).

% fun_upd_restrict
tff(fact_6179_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_6180_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xb: A] : restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))) = fun_upd(A,option(B),F2,Xb,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_6181_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_6182_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)),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aTP_Lamp_abb(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_6183_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_6184_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_6185_map__upds__apply__nontin,axiom,
    ! [B: $tType,A: $tType,Xb: A,Xs: list(A),F2: 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,F2,Xs,Ys),Xb) = aa(A,option(B),F2,Xb) ) ) ).

% map_upds_apply_nontin
tff(fact_6186_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_6187_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ia: nat,Mb: fun(A,option(B)),Ys: list(B),Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Ia)
     => ( map_upds(A,B,Mb,Xs,list_update(B,Ys,Ia,Y)) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_6188_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_6189_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_6190_union__comp__emptyL,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),C5: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A3,C5) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,B3,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))),A3),B3),C5) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyL
tff(fact_6191_union__comp__emptyR,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),B3: set(product_prod(A,A)),C5: set(product_prod(A,A))] :
      ( ( relcomp(A,A,A,A3,B3) = bot_bot(set(product_prod(A,A))) )
     => ( ( relcomp(A,A,A,A3,C5) = bot_bot(set(product_prod(A,A))) )
       => ( relcomp(A,A,A,A3,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),B3),C5)) = bot_bot(set(product_prod(A,A))) ) ) ) ).

% union_comp_emptyR
tff(fact_6192_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)))),insert2(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R)) ) ).

% min_ext_compat
tff(fact_6193_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_aaw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_aay(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_6194_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)))),insert2(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R)) ) ).

% max_ext_compat
tff(fact_6195_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = collect(A,aa(set(nat),fun(A,$o),aTP_Lamp_abc(list(A),fun(set(nat),fun(A,$o)),Xs),I5)) ).

% set_nths
tff(fact_6196_nths__nil,axiom,
    ! [A: $tType,A3: set(nat)] : nths(A,nil(A),A3) = nil(A) ).

% nths_nil
tff(fact_6197_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_6198_distinct__nthsI,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => distinct(A,nths(A,Xs,I5)) ) ).

% distinct_nthsI
tff(fact_6199_in__set__nthsD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I5: set(nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),nths(A,Xs,I5)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_nthsD
tff(fact_6200_notin__set__nthsI,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I5: 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,I5))) ) ).

% notin_set_nthsI
tff(fact_6201_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I5))),aa(list(A),set(A),set2(A),Xs)) ).

% set_nths_subset
tff(fact_6202_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => member(nat,I2,I5) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_6203_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I5)) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(set(nat),fun(nat,$o),aTP_Lamp_abd(list(A),fun(set(nat),fun(nat,$o)),Xs),I5))) ).

% length_nths
tff(fact_6204_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A1),A22),max_ext(A,R))
     => ~ ( aa(set(A),$o,finite_finite2(A),A1)
         => ( aa(set(A),$o,finite_finite2(A),A22)
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X: A] :
                    ( member(A,X,A1)
                   => ? [Xa2: A] :
                        ( member(A,Xa2,A22)
                        & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Xa2),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_6205_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A1),A22),max_ext(A,R))
    <=> ( aa(set(A),$o,finite_finite2(A),A1)
        & aa(set(A),$o,finite_finite2(A),A22)
        & ( A22 != bot_bot(set(A)) )
        & ! [X2: A] :
            ( member(A,X2,A1)
           => ? [Xa3: A] :
                ( member(A,Xa3,A22)
                & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X2),Xa3),R) ) ) ) ) ).

% max_ext.simps
tff(fact_6206_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),product_Pair(A,A,X3),Xa),R) ) )
           => member(product_prod(set(A),set(A)),aa(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_6207_minus__fold__remove,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B3),A3) = finite_fold(A,set(A),remove(A),B3,A3) ) ) ).

% minus_fold_remove
tff(fact_6208_finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( finite5375528669736107172at_top(A,A3) = principal(set(A),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A3),bot_bot(set(set(A))))) ) ) ).

% finite_subsets_at_top_finite
tff(fact_6209_member__remove,axiom,
    ! [A: $tType,Xb: A,Y: A,A3: set(A)] :
      ( member(A,Xb,aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Y),A3))
    <=> ( member(A,Xb,A3)
        & ( Xb != Y ) ) ) ).

% member_remove
tff(fact_6210_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( ! [X6: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X6)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),A3)
           => aa(set(A),$o,P,X6) ) )
     => eventually(set(A),P,finite5375528669736107172at_top(A,A3)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_6211_eventually__finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( eventually(set(A),P,finite5375528669736107172at_top(A,A3))
      <=> aa(set(A),$o,P,A3) ) ) ).

% eventually_finite_subsets_at_top_finite
tff(fact_6212_finite__subsets__at__top__neq__bot,axiom,
    ! [A: $tType,A3: set(A)] : finite5375528669736107172at_top(A,A3) != bot_bot(filter(set(A))) ).

% finite_subsets_at_top_neq_bot
tff(fact_6213_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,P: fun(set(A),$o),A3: set(A)] :
      ( eventually(set(A),P,finite5375528669736107172at_top(A,A3))
    <=> ? [X8: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X8)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X8),A3)
          & ! [Y6: set(A)] :
              ( ( aa(set(A),$o,finite_finite2(A),Y6)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X8),Y6)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Y6),A3) )
             => aa(set(A),$o,P,Y6) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_6214_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A3: set(A)] : finite5375528669736107172at_top(A,A3) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image2(set(A),filter(set(A)),aTP_Lamp_abf(set(A),fun(set(A),filter(set(A))),A3)),collect(set(A),aTP_Lamp_abg(set(A),fun(set(A),$o),A3)))) ).

% finite_subsets_at_top_def
tff(fact_6215_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_6216_remove__def,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xb),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) ).

% remove_def
tff(fact_6217_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,set(B)),A3: set(B),F4: filter(A)] :
      ( filterlim(A,set(B),F2,finite5375528669736107172at_top(B,A3),F4)
    <=> ! [X8: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),X8)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X8),A3) )
         => eventually(A,aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_abh(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),F2),A3),X8),F4) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_6218_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_6219_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_abi(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_6220_less__rat,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A2,B2)),fract(C2,D2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))) ) ) ) ).

% less_rat
tff(fact_6221_Rat__induct__pos,axiom,
    ! [P: 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,P,fract(A4,B4)) )
     => aa(rat,$o,P,Q2) ) ).

% Rat_induct_pos
tff(fact_6222_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_6223_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_6224_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_6225_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_6226_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_6227_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_6228_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_6229_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_6230_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_6231_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_abj(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Sup_fin.eq_fold'
tff(fact_6232_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_abk(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Inf_fin.eq_fold'
tff(fact_6233_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = A2 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_6234_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)),insert2(A),Xb),bot_bot(set(A)))) = Xb ) ).

% Sup_fin.singleton
tff(fact_6235_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)),insert2(A),Xb),bot_bot(set(A)))) = Xb ) ).

% Inf_fin.singleton
tff(fact_6236_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A2) = A2 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_6237_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_6238_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_6239_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_6240_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_6241_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_6242_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A2) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_6243_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A3) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_6244_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_6245_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3))
             => ! [A9: A] :
                  ( member(A,A9,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A9) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_6246_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A3)
                 => 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),A3)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_6247_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),Xb)
             => ! [A9: A] :
                  ( member(A,A9,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),Xb) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_6248_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A3)
                 => 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),A3)),Xb) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_6249_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3))
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X2) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_6250_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),Xb)
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xb) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_6251_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_6252_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_6253_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_6254_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_6255_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_6256_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_6257_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_6258_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_6259_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( aa(set(A),$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_6260_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( aa(set(A),$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_6261_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_6262_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A3) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_6263_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A3),A3) ) ) ) ) ).

% Inf_fin.closed
tff(fact_6264_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_6265_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A3),A3) ) ) ) ) ).

% Sup_fin.closed
tff(fact_6266_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_6267_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_6268_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_6269_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = finite_fold(A,A,inf_inf(A),Xb,A3) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_6270_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = finite_fold(A,A,sup_sup(A),Xb,A3) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_6271_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_abl(set(A),fun(A,fun(A,$o)),A3),Xb))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_6272_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),collect(A,aa(set(A),fun(A,$o),aTP_Lamp_abm(set(A),fun(set(A),fun(A,$o)),A3),B3))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_6273_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_abn(set(A),fun(A,fun(A,$o)),A3),Xb))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_6274_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),collect(A,aa(set(A),fun(A,$o),aTP_Lamp_abo(set(A),fun(set(A),fun(A,$o)),A3),B3))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_6275_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_6276_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_6277_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_6278_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_6279_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_6280_pos__deriv__imp__strict__mono,axiom,
    ! [F2: fun(real,real),F9: fun(real,real)] :
      ( ! [X3: real] : has_field_derivative(real,F2,aa(real,real,F9,X3),topolo174197925503356063within(real,X3,top_top(set(real))))
     => ( ! [X3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F9,X3))
       => order_strict_mono(real,real,F2) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_6281_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_6282_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% strict_mono_less_eq
tff(fact_6283_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_6284_strict__mono__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
         => order_mono(A,B,F2) ) ) ).

% strict_mono_mono
tff(fact_6285_strict__mono__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( ( aa(A,B,F2,Xb) = aa(A,B,F2,Y) )
          <=> ( Xb = Y ) ) ) ) ).

% strict_mono_eq
tff(fact_6286_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% strict_monoD
tff(fact_6287_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y3)) )
         => order_strict_mono(A,B,F2) ) ) ).

% strict_monoI
tff(fact_6288_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
        <=> ! [X2: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) ) ) ) ).

% strict_mono_def
tff(fact_6289_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% strict_mono_less
tff(fact_6290_strict__mono__add,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A] : order_strict_mono(A,A,aTP_Lamp_lm(A,fun(A,A),K)) ) ).

% strict_mono_add
tff(fact_6291_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_6292_ring__1__class_Oof__int__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),product_case_prod(nat,nat,A,aTP_Lamp_kt(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_6293_sorted__key__list__of__set__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B)] : linord144544945434240204of_set(A,B,F2) = finite_folding_F(A,list(A),linorder_insort_key(A,B,F2),nil(A)) ) ).

% sorted_key_list_of_set_def
tff(fact_6294_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_6295_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_6296_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_6297_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_6298_strict__mono__imp__increasing,axiom,
    ! [F2: fun(nat,nat),Nb: nat] :
      ( order_strict_mono(nat,nat,F2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,F2,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_6299_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_6300_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($o))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_kv(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int_def
tff(fact_6301_folding__on_OF_Ocong,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B] : finite_folding_F(A,B,F2,Z) = finite_folding_F(A,B,F2,Z) ).

% folding_on.F.cong
tff(fact_6302_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F2) = id(A) ).

% funpow_simps_right(1)
tff(fact_6303_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($o))),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o),aTP_Lamp_kx(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int_def
tff(fact_6304_nat__def,axiom,
    nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))) ).

% nat_def
tff(fact_6305_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_strict_mono(nat,A,F2)
        <=> ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2))) ) ) ).

% strict_mono_Suc_iff
tff(fact_6306_summable__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ member(nat,N,aa(set(nat),set(nat),image2(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abp(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2))
            <=> summable(A,F2) ) ) ) ) ).

% summable_mono_reindex
tff(fact_6307_sums__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A),C2: A] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ member(nat,N,aa(set(nat),set(nat),image2(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abp(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2),C2)
            <=> sums(A,F2,C2) ) ) ) ) ).

% sums_mono_reindex
tff(fact_6308_suminf__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ member(nat,N,aa(set(nat),set(nat),image2(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abq(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2)) = suminf(A,F2) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_6309_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( ! [X3: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Y3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,X3))),real_V7770717601297561774m_norm(A,aa(nat,A,F2,Y3))) )
         => ( order_strict_mono(nat,nat,G)
           => ( bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_abr(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat))
            <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ) ).

% increasing_Bseq_subseq_iff
tff(fact_6310_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Xb: A,Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A3) = aa(B,B,aa(A,fun(B,B),F2,Xb),aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_6311_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(A,fun(B,B),F2,Xb),aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_6312_folding__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,S2)
         => ( member(A,Y3,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y3)) ) ) )
     => finite_folding_on(A,B,S2,F2) ) ).

% folding_on.intro
tff(fact_6313_folding__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,Y: A] :
      ( finite_folding_on(A,B,S2,F2)
     => ( member(A,Xb,S2)
       => ( member(A,Y,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),aa(A,fun(B,B),F2,Xb)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xb)),aa(A,fun(B,B),F2,Y)) ) ) ) ) ).

% folding_on.comp_fun_commute_on
tff(fact_6314_folding__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S2,F2)
    <=> ! [X2: A,Y4: A] :
          ( member(A,X2,S2)
         => ( member(A,Y4,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),aa(A,fun(B,B),F2,X2)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X2)),aa(A,fun(B,B),F2,Y4)) ) ) ) ) ).

% folding_on_def
tff(fact_6315_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_aal(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_6316_folding__on_Oempty,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),bot_bot(set(A))) = Z ) ) ).

% folding_on.empty
tff(fact_6317_folding__on_Oinfinite,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( ~ aa(set(A),$o,finite_finite2(A),A3)
       => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A3) = Z ) ) ) ).

% folding_on.infinite
tff(fact_6318_folding__on_Oeq__fold,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Z: B,A3: set(A)] :
      ( finite_folding_on(A,B,S2,F2)
     => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A3) = finite_fold(A,B,F2,Z,A3) ) ) ).

% folding_on.eq_fold
tff(fact_6319_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_xr(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_6320_card__def,axiom,
    ! [A: $tType] : finite_card(A) = finite_folding_F(A,nat,aTP_Lamp_aal(A,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_6321_folding__on_Oinsert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(A,fun(B,B),F2,Xb),aa(set(A),B,finite_folding_F(A,B,F2,Z),A3)) ) ) ) ) ) ).

% folding_on.insert
tff(fact_6322_folding__idem__on_Oinsert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(A,fun(B,B),F2,Xb),aa(set(A),B,finite_folding_F(A,B,F2,Z),A3)) ) ) ) ) ).

% folding_idem_on.insert_idem
tff(fact_6323_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_6324_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_6325_folding__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,Y: A] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
     => ( member(A,Xb,S2)
       => ( member(A,Y,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xb)),aa(A,fun(B,B),F2,Xb)) = aa(A,fun(B,B),F2,Xb) ) ) ) ) ).

% folding_idem_on.comp_fun_idem_on
tff(fact_6326_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_6327_folding__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
     => finite_folding_on(A,B,S2,F2) ) ).

% folding_idem_on.axioms(1)
tff(fact_6328_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_6329_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_6330_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_6331_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_6332_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_6333_folding__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
    <=> ( finite_folding_on(A,B,S2,F2)
        & finite6916993218817215295axioms(A,B,S2,F2) ) ) ).

% folding_idem_on_def
tff(fact_6334_folding__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S2,F2)
     => ( finite6916993218817215295axioms(A,B,S2,F2)
       => finite1890593828518410140dem_on(A,B,S2,F2) ) ) ).

% folding_idem_on.intro
tff(fact_6335_folding__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite6916993218817215295axioms(A,B,S2,F2)
    <=> ! [X2: A,Y4: A] :
          ( member(A,X2,S2)
         => ( member(A,Y4,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X2)),aa(A,fun(B,B),F2,X2)) = aa(A,fun(B,B),F2,X2) ) ) ) ) ).

% folding_idem_on_axioms_def
tff(fact_6336_folding__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,S2)
         => ( member(A,Y3,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,X3)) = aa(A,fun(B,B),F2,X3) ) ) )
     => finite6916993218817215295axioms(A,B,S2,F2) ) ).

% folding_idem_on_axioms.intro
tff(fact_6337_folding__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S2,F2)
     => finite6916993218817215295axioms(A,B,S2,F2) ) ).

% folding_idem_on.axioms(2)
tff(fact_6338_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_6339_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xb),Z),A3) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_6340_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_6341_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_6342_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_6343_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) ) ).

% diff_numeral_special(5)
tff(fact_6344_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(Mb)) ) ).

% diff_numeral_special(6)
tff(fact_6345_num__induct,axiom,
    ! [P: fun(num,$o),Xb: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X3: num] :
            ( aa(num,$o,P,X3)
           => aa(num,$o,P,inc(X3)) )
       => aa(num,$o,P,Xb) ) ) ).

% num_induct
tff(fact_6346_comp__fun__idem__on_Ofun__left__idem,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( member(A,Xb,S2)
       => ( aa(B,B,aa(A,fun(B,B),F2,Xb),aa(B,B,aa(A,fun(B,B),F2,Xb),Z)) = aa(B,B,aa(A,fun(B,B),F2,Xb),Z) ) ) ) ).

% comp_fun_idem_on.fun_left_idem
tff(fact_6347_add__inc,axiom,
    ! [Xb: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y)) ).

% add_inc
tff(fact_6348_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( member(A,Xb,S2)
       => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xb)),aa(A,fun(B,B),F2,Xb)) = aa(A,fun(B,B),F2,Xb) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_6349_comp__fun__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => finite4664212375090638736ute_on(A,B,S2,F2) ) ).

% comp_fun_idem_on.axioms(1)
tff(fact_6350_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_6351_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : inc(bit0(Xb)) = bit1(Xb) ).

% inc.simps(2)
tff(fact_6352_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : inc(bit1(Xb)) = bit0(inc(Xb)) ).

% inc.simps(3)
tff(fact_6353_add__One,axiom,
    ! [Xb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2) = inc(Xb) ).

% add_One
tff(fact_6354_inc__BitM__eq,axiom,
    ! [Nb: num] : inc(bitM(Nb)) = bit0(Nb) ).

% inc_BitM_eq
tff(fact_6355_BitM__inc__eq,axiom,
    ! [Nb: num] : bitM(inc(Nb)) = bit1(Nb) ).

% BitM_inc_eq
tff(fact_6356_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),Xb),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),Xb),Y)),Xb) ).

% mult_inc
tff(fact_6357_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_6358_sub__inc__One__eq,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : neg_numeral_sub(A,inc(Nb),one2) = aa(num,A,numeral_numeral(A),Nb) ) ).

% sub_inc_One_eq
tff(fact_6359_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F2: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G),top_top(set(C)))),S2)
       => finite673082921795544331dem_on(C,B,R,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F2),G)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_6360_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(B,B,aa(A,fun(B,B),F2,Xb),finite_fold(A,B,F2,Z,A3)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_6361_positive__rat,axiom,
    ! [A2: int,B2: int] :
      ( 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_6362_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_6363_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_6364_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_6365_take0,axiom,
    ! [A: $tType,X: list(A)] : take(A,zero_zero(nat),X) = nil(A) ).

% take0
tff(fact_6366_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_6367_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_6368_nth__take,axiom,
    ! [A: $tType,Ia: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),Nb)
     => ( aa(nat,A,nth(A,take(A,Nb,Xs)),Ia) = aa(nat,A,nth(A,Xs),Ia) ) ) ).

% nth_take
tff(fact_6369_take__update__cancel,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( take(A,Nb,list_update(A,Xs,Mb,Y)) = take(A,Nb,Xs) ) ) ).

% take_update_cancel
tff(fact_6370_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_6371_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_6372_take__Nil,axiom,
    ! [A: $tType,Nb: nat] : take(A,Nb,nil(A)) = nil(A) ).

% take_Nil
tff(fact_6373_distinct__take,axiom,
    ! [A: $tType,Xs: list(A),Ia: nat] :
      ( distinct(A,Xs)
     => distinct(A,take(A,Ia,Xs)) ) ).

% distinct_take
tff(fact_6374_take__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ! [I2: nat] : take(A,I2,Xs) = take(A,I2,Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_6375_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_6376_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_6377_less__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xb),Y)
    <=> positive(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Y),Xb)) ) ).

% less_rat_def
tff(fact_6378_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_6379_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_6380_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys))
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
         => ( take(A,K,Xs) = take(A,K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_6381_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      map_upds(A,B,fun_upd(A,option(B),F2,Xb,aa(B,option(B),some(B),Y)),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,F2,Xs,Ys),fun_upd(A,option(B),map_upds(A,B,F2,Xs,Ys),Xb,aa(B,option(B),some(B),Y))) ).

% map_upd_upds_conv_if
tff(fact_6382_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys),lex(A,R2))
     => ~ ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ys))
             => ( ( take(A,I2,Xs) = take(A,I2,Ys) )
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Ys),I2)),R2) ) ) ) ) ).

% lex_take_index
tff(fact_6383_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_tx(A,fun(A,$o))) = ord_max(A) ) ) ).

% dual_min
tff(fact_6384_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)),product_Pair(list(A),list(A),Xs),nil(A)),lex(A,R2)) ).

% Nil2_notin_lex
tff(fact_6385_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)),product_Pair(list(A),list(A),nil(A)),Ys),lex(A,R2)) ).

% Nil_notin_lex
tff(fact_6386_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_6387_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_6388_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys),listrel1(A,R2))
    <=> ? [Y4: A,N2: nat] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),N2)),Y4),R2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
          & ( Ys = list_update(A,Xs,N2,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_6389_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_abs(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lenlex_conv
tff(fact_6390_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)),product_Pair(list(A),list(A),nil(A)),Ns),lenlex(A,R2))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_6391_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)),product_Pair(list(A),list(A),nil(A)),Xs),listrel1(A,R2)) ).

% not_Nil_listrel1
tff(fact_6392_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)),product_Pair(list(A),list(A),Xs),nil(A)),listrel1(A,R2)) ).

% not_listrel1_Nil
tff(fact_6393_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)),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_6394_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xb),Y),transitive_rtrancl(list(A),listrel1(A,R2)))
     => ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_6395_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_6396_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_6397_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),product_Pair(A,A,X3),X3),R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Xs),lenlex(A,R2)) ) ).

% lenlex_irreflexive
tff(fact_6398_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)),product_Pair(list(A),list(A),Ns),nil(A)),lenlex(A,R2)) ).

% Nil_lenlex_iff2
tff(fact_6399_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ms),Ns),lenlex(A,R2))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_6400_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)),product_Pair(list(A),list(A),Xs),Ys),listrel1(A,collect(product_prod(A,A),product_case_prod(A,A,$o,R2)))) ) ).

% listrel1p_def
tff(fact_6401_card__Min__le__sum,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => 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),A3)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image2(A,nat,F2),A3)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A3)) ) ).

% card_Min_le_sum
tff(fact_6402_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)),insert2(A),Xb),bot_bot(set(A)))) = Xb ) ).

% Min_singleton
tff(fact_6403_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A3))
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X2) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6404_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A3))
            <=> ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X2) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_6405_Min__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A3: set(A),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_qf(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Min_const
tff(fact_6406_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_6407_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_6408_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A3),A3) ) ) ) ).

% Min_in
tff(fact_6409_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,A2,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A2) ) ) ) ).

% Min.coboundedI
tff(fact_6410_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [Y3: A] :
                ( member(A,Y3,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y3) )
           => ( member(A,Xb,A3)
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = Xb ) ) ) ) ) ).

% Min_eqI
tff(fact_6411_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),Xb) ) ) ) ).

% Min_le
tff(fact_6412_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_6413_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = Mb )
            <=> ( member(A,Mb,A3)
                & ! [X2: A] :
                    ( member(A,X2,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X2) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6414_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),Xb)
            <=> ? [X2: A] :
                  ( member(A,X2,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xb) ) ) ) ) ) ).

% Min_le_iff
tff(fact_6415_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798350308766er_Min(A),A3) )
            <=> ( member(A,Mb,A3)
                & ! [X2: A] :
                    ( member(A,X2,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X2) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6416_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A3))
             => ! [A9: A] :
                  ( member(A,A9,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A9) ) ) ) ) ) ).

% Min.boundedE
tff(fact_6417_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A3)
                 => 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),A3)) ) ) ) ) ).

% Min.boundedI
tff(fact_6418_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),Xb)
            <=> ? [X2: A] :
                  ( member(A,X2,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Xb) ) ) ) ) ) ).

% Min_less_iff
tff(fact_6419_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [B4: A] :
                ( member(A,B4,A3)
               => 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)),insert2(A),A2),A3)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_6420_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ) ) ) ).

% Min_Inf
tff(fact_6421_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_6422_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_6423_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B3)),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) ) ) ) ) ).

% Min.subset_imp
tff(fact_6424_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_6425_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( aa(set(A),$o,finite_finite2(A),A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(A,B,F2,aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_6426_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S2: set(A),F2: fun(A,B),K: B] :
          ( aa(set(A),$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_ru(fun(A,B),fun(B,fun(A,B)),F2),K)),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,F2),S2))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_6427_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_tx(A,fun(A,$o))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_6428_f__arg__min__list__f,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( aa(A,B,F2,arg_min_list(A,B,F2,Xs)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F2),aa(list(A),set(A),set2(A),Xs))) ) ) ) ).

% f_arg_min_list_f
tff(fact_6429_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_6430_arg__min__list__in,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B)] :
          ( ( Xs != nil(A) )
         => member(A,arg_min_list(A,B,F2,Xs),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% arg_min_list_in
tff(fact_6431_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_abt(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Min.eq_fold'
tff(fact_6432_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_6433_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_6434_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_6435_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_6436_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_6437_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_6438_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_6439_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
        <=> ( aa(A,$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),C2) ) ) ) ).

% min.bounded_iff
tff(fact_6440_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_6441_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_6442_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_6443_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_6444_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_6445_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_6446_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_6447_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_6448_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: 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),Y)) = Xb ) ).

% max_min_same(1)
tff(fact_6449_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),Xb) = Xb ) ).

% max_min_same(2)
tff(fact_6450_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),Y) = Y ) ).

% max_min_same(3)
tff(fact_6451_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)) = Y ) ).

% max_min_same(4)
tff(fact_6452_take__replicate,axiom,
    ! [A: $tType,Ia: nat,K: nat,Xb: A] : take(A,Ia,replicate(A,K,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ia),K),Xb) ).

% take_replicate
tff(fact_6453_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V2)) ) ).

% min_number_of(1)
tff(fact_6454_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_6455_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_6456_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_6457_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_6458_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_6459_min__0__1_I6_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)),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_6460_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),Nb)) ).

% min_numeral_Suc
tff(fact_6461_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_6462_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_6463_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ).

% min_number_of(2)
tff(fact_6464_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) ) ).

% min_number_of(3)
tff(fact_6465_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ).

% min_number_of(4)
tff(fact_6466_Int__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_atLeastAtMost
tff(fact_6467_Int__atLeastAtMostL1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,D2: 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),D2)) = set_or1337092689740270186AtMost(A,A2,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_atLeastAtMostL1
tff(fact_6468_Int__atLeastAtMostR1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,D2: 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,C2,D2)) = set_or1337092689740270186AtMost(A,C2,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_atLeastAtMostR1
tff(fact_6469_Int__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_atLeastLessThan
tff(fact_6470_Int__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D2)) = set_or5935395276787703475ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_greaterThanLessThan
tff(fact_6471_Int__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_greaterThanAtMost
tff(fact_6472_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) ) ) ) ) ).

% Min_insert
tff(fact_6473_Min_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(A),A,lattic643756798350308766er_Min(A),A3) ) ) ) ) ).

% Min.in_idem
tff(fact_6474_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_6475_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_6476_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_6477_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_6478_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_6479_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_6480_of__int__min,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,Y: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_min(int),Xb),Y)) = 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),Y)) ) ).

% of_int_min
tff(fact_6481_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_6482_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_6483_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Xb),Y)) = 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),Y)) ) ).

% of_nat_min
tff(fact_6484_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: 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),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ).

% min_add_distrib_right
tff(fact_6485_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),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),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_6486_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xb: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_6487_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xb: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_6488_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),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),Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_6489_min__diff,axiom,
    ! [Mb: nat,Ia: 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),Ia)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Ia)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),Ia) ).

% min_diff
tff(fact_6490_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$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),C2),D2)) ) ) ) ).

% min.mono
tff(fact_6491_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_6492_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_6493_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))
         => ~ ( aa(A,$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),C2) ) ) ) ).

% min.boundedE
tff(fact_6494_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ) ) ).

% min.boundedI
tff(fact_6495_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_6496_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_6497_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_6498_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_6499_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_6500_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.coboundedI1
tff(fact_6501_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2) ) ) ).

% min.coboundedI2
tff(fact_6502_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_6503_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_6504_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y) = Xb ) ) ) ).

% min_absorb1
tff(fact_6505_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_6506_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_6507_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xb: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% max_mult_distrib_left
tff(fact_6508_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xb: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% min_mult_distrib_left
tff(fact_6509_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% max_mult_distrib_right
tff(fact_6510_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% min_mult_distrib_right
tff(fact_6511_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P3: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xb,P3)),divide_divide(A,Y,P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xb,P3)),divide_divide(A,Y,P3))) ) ).

% max_divide_distrib_right
tff(fact_6512_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P3: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xb,P3)),divide_divide(A,Y,P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xb,P3)),divide_divide(A,Y,P3))) ) ).

% min_divide_distrib_right
tff(fact_6513_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_abu(nat,fun(nat,nat),Nb),Mb) ).

% min_Suc2
tff(fact_6514_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_abv(nat,fun(nat,nat),Nb),Mb) ).

% min_Suc1
tff(fact_6515_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)),insert2(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_6516_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y3: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X3)),aa(A,A,H,Y3))
         => ( aa(set(A),$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_6517_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( B3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B3)),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(A),A,lattic643756798350308766er_Min(A),A3) ) ) ) ) ) ).

% Min.subset
tff(fact_6518_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A3),A3) ) ) ) ) ).

% Min.closed
tff(fact_6519_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ~ member(A,Xb,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_6520_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B3)
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),aa(set(A),A,lattic643756798350308766er_Min(A),B3)) ) ) ) ) ) ) ).

% Min.union
tff(fact_6521_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = finite_fold(A,A,ord_min(A),Xb,A3) ) ) ) ).

% Min.eq_fold
tff(fact_6522_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))))) ) ) ) ).

% Min.insert_remove
tff(fact_6523_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),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)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Min.remove
tff(fact_6524_lexord__take__index__conv,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xb),Y),lexord(A,R2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xb)),aa(list(A),nat,size_size(list(A)),Y))
          & ( take(A,aa(list(A),nat,size_size(list(A)),Xb),Y) = Xb ) )
        | ? [I4: nat] :
            ( aa(nat,$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)),Y)))
            & ( take(A,I4,Xb) = take(A,I4,Y) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xb),I4)),aa(nat,A,nth(A,Y),I4)),R2) ) ) ) ).

% lexord_take_index_conv
tff(fact_6525_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_tx(A,fun(A,$o))) = ord_min(A) ) ) ).

% dual_max
tff(fact_6526_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_6527_inf__int__def,axiom,
    inf_inf(int) = ord_min(int) ).

% inf_int_def
tff(fact_6528_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_6529_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_6530_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),product_Pair(A,A,X3),X3),R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Xs),lexord(A,R2)) ) ).

% lexord_irreflexive
tff(fact_6531_lexord__linear,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: list(A),Y: list(A)] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(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),product_Pair(A,A,B4),A4),R2) )
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xb),Y),lexord(A,R2))
        | ( Xb = Y )
        | member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Y),Xb),lexord(A,R2)) ) ) ).

% lexord_linear
tff(fact_6532_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)),product_Pair(list(A),list(A),Xb),nil(A)),lexord(A,R2)) ).

% lexord_Nil_right
tff(fact_6533_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A,Y3: 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),product_Pair(A,A,X3),Y3),R2)
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),Z2),R2)
             => member(product_prod(A,A),aa(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)),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)),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)),product_Pair(list(A),list(A),Xs),Zs),lexord(A,R2)) ) ) ) ).

% lexord_partial_trans
tff(fact_6534_lexord__lex,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xb),Y),lex(A,R2))
    <=> ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xb),Y),lexord(A,R2))
        & ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_6535_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)),product_Pair(list(A),list(A),Xs),Ys),lexord(A,collect(product_prod(A,A),product_case_prod(A,A,$o,R2)))) ) ).

% List.lexordp_def
tff(fact_6536_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = collect(product_prod(A,B),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_abw(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys)) ).

% set_zip
tff(fact_6537_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_6538_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_6539_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_6540_zip__replicate,axiom,
    ! [A: $tType,B: $tType,Ia: nat,Xb: A,J: nat,Y: B] : zip(A,B,replicate(A,Ia,Xb),replicate(B,J,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Ia),J),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y)) ).

% zip_replicate
tff(fact_6541_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_6542_nth__zip,axiom,
    ! [A: $tType,B: $tType,Ia: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),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)),Ia) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),Ia)),aa(nat,B,nth(B,Ys),Ia)) ) ) ) ).

% nth_zip
tff(fact_6543_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_6544_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_6545_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ia: nat,Xb: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,Ia,Xb),list_update(B,Ys,Ia,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),Ia,aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y)) ).

% zip_update
tff(fact_6546_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_6547_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_6548_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(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)))
     => member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ).

% set_zip_rightD
tff(fact_6549_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(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)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_zip_leftD
tff(fact_6550_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(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)))
     => ~ ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ~ member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ) ).

% in_set_zipE
tff(fact_6551_zip__same,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( member(product_prod(A,A),aa(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_6552_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(product_prod(A,B)),$o)] :
      ( ! [Zs2: list(A),Ws: list(B),N: nat] :
          ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws) )
         => ( ( N = 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,N,Xs) )
             => ( ( Ws = take(B,N,Ys) )
               => aa(list(product_prod(A,B)),$o,P,zip(A,B,Zs2,Ws)) ) ) ) )
     => aa(list(product_prod(A,B)),$o,P,zip(A,B,Xs,Ys)) ) ).

% zip_obtain_same_length
tff(fact_6553_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X2: product_prod(A,A)] :
            ( member(product_prod(A,A),X2,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys)))
           => aa(product_prod(A,A),$o,product_case_prod(A,A,$o,fequal(A)),X2) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_6554_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(B,Y,aa(list(B),set(B),set2(B),Ys))
       => ~ ! [X3: A] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_6555_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))
       => ~ ! [Y3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y3),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_6556_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,product_case_prod(list(A),list(A),$o,aTP_Lamp_abx(list(A),fun(list(A),$o))),X3) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_6557_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,product_case_prod(list(A),list(A),$o,aTP_Lamp_abx(list(A),fun(list(A),$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_6558_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X2: product_prod(A,B)] :
            ( member(product_prod(A,B),X2,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
           => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,aTP_Lamp_aby(set(product_prod(A,B)),fun(A,fun(B,$o)),R2)),X2) ) ) ) ).

% listrel_iff_zip
tff(fact_6559_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Ia: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),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),Ia)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),Ia)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_6560_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)),product_Pair(list(A),list(A),Xs),Xs),listrel(A,A,transitive_rtrancl(A,R2))) ).

% listrel_rtrancl_refl
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_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)),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_6563_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)),product_Pair(list(A),list(B),nil(A)),nil(B)),listrel(A,B,R2)) ).

% listrel.Nil
tff(fact_6564_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)),product_Pair(list(A),list(B),nil(A)),Xs),listrel(A,B,R2))
     => ( Xs = nil(B) ) ) ).

% listrel_Nil1
tff(fact_6565_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)),product_Pair(list(A),list(B),Xs),nil(B)),listrel(A,B,R2))
     => ( Xs = nil(A) ) ) ).

% listrel_Nil2
tff(fact_6566_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)),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)),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)),product_Pair(list(A),list(A),Xs),Zs),listrel(A,A,transitive_rtrancl(A,R2))) ) ) ).

% listrel_rtrancl_trans
tff(fact_6567_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_6568_map__of__zip__inject,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xs) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xs) )
       => ( distinct(B,Xs)
         => ( ( map_of(B,A,zip(B,A,Xs,Ys)) = map_of(B,A,zip(B,A,Xs,Zs)) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
tff(fact_6569_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)),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)),product_Pair(list(A),list(A),Xs),Ys),listrel(A,A,transitive_rtrancl(A,R2))) ) ).

% listrel_reflcl_if_listrel1
tff(fact_6570_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_6571_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_6572_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)),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)),product_Pair(list(A),list(A),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R2))) ) ).

% rtrancl_listrel1_if_listrel
tff(fact_6573_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_6574_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_6575_map__of__zip__upd,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A),Xb: B,Y: 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),Y)) = 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_6576_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),Xs))
           => member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),N2)),aa(nat,B,nth(B,Ys),N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_6577_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_6578_listrel__def,axiom,
    ! [B: $tType,A: $tType,X: set(product_prod(A,B))] : listrel(A,B,X) = collect(product_prod(list(A),list(B)),product_case_prod(list(A),list(B),$o,listrelp(A,B,aTP_Lamp_aby(set(product_prod(A,B)),fun(A,fun(B,$o)),X)))) ).

% listrel_def
tff(fact_6579_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_abz(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_6580_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_6581_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),$o),listrelp(A,B,aTP_Lamp_aby(set(product_prod(A,B)),fun(A,fun(B,$o)),R2)),X),Xa)
    <=> member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),X),Xa),listrel(A,B,R2)) ) ).

% listrelp_listrel_eq
tff(fact_6582_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),Xb: B,Y: A,Z: A] :
      ( ( aa(B,option(A),Mb,Xb) = aa(A,option(A),some(A),Y) )
     => ( 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)),insert2(A),Y),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_6583_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_6584_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( ( dom(A,B,F2) = bot_bot(set(A)) )
    <=> ! [X2: A] : aa(A,option(B),F2,X2) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_6585_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_aca(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_6586_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_6587_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),Xb: A,Y: option(B)] :
      dom(A,B,fun_upd(A,option(B),F2,Xb,Y)) = $ite(Y = none(B),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),dom(A,B,F2))) ).

% dom_fun_upd
tff(fact_6588_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_6589_finite__ran,axiom,
    ! [B: $tType,A: $tType,P3: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,P3))
     => aa(set(B),$o,finite_finite2(B),ran(A,B,P3)) ) ).

% finite_ran
tff(fact_6590_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_6591_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_6592_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,F2))
     => ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
       => ? [X3: A] : aa(A,option(B),F2,X3) = none(B) ) ) ).

% finite_map_freshness
tff(fact_6593_finite__set__of__finite__maps,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => aa(set(fun(A,option(B))),$o,finite_finite2(fun(A,option(B))),collect(fun(A,option(B)),aa(set(B),fun(fun(A,option(B)),$o),aTP_Lamp_acb(set(A),fun(set(B),fun(fun(A,option(B)),$o)),A3),B3))) ) ) ).

% finite_set_of_finite_maps
tff(fact_6594_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),P: fun(fun(A,option(B)),$o)] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,Mb))
     => ( aa(fun(A,option(B)),$o,P,aTP_Lamp_aca(A,option(B)))
       => ( ! [K3: A,V3: B,M2: fun(A,option(B))] :
              ( aa(set(A),$o,finite_finite2(A),dom(A,B,M2))
             => ( ~ member(A,K3,dom(A,B,M2))
               => ( aa(fun(A,option(B)),$o,P,M2)
                 => aa(fun(A,option(B)),$o,P,fun_upd(A,option(B),M2,K3,aa(B,option(B),some(B),V3))) ) ) )
         => aa(fun(A,option(B)),$o,P,Mb) ) ) ) ).

% finite_Map_induct
tff(fact_6595_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xb: A] :
      ( ( dom(A,B,F2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))) )
    <=> ? [V7: B] : F2 = fun_upd(A,option(B),aTP_Lamp_aca(A,option(B)),Xb,aa(B,option(B),some(B),V7)) ) ).

% dom_eq_singleton_conv
tff(fact_6596_refl__on__singleton,axiom,
    ! [A: $tType,Xb: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(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))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_6597_total__on__singleton,axiom,
    ! [A: $tType,Xb: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(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))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_6598_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_6599_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_6600_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_6601_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_6602_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),product_Pair(A,B,aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys)) ) ) ) ).

% hd_zip
tff(fact_6603_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_6604_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_6605_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_6606_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_6607_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_6608_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_6609_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_6610_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] :
      aa(list(A),nat,size_list(A,F2),Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F2),aa(list(A),list(A),tl(A),Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_6611_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_6612_rotate0,axiom,
    ! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).

% rotate0
tff(fact_6613_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_6614_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_6615_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_6616_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_6617_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_6618_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_6619_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_6620_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_6621_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_6622_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_6623_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_6624_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_6625_list_Osel_I2_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).

% list.sel(2)
tff(fact_6626_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_6627_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_6628_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_6629_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_6630_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_6631_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_6632_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_6633_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_6634_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_6635_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_6636_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Xb: A] :
      ( ( find(A,P,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,$o,P,aa(nat,A,nth(A,Xs),I4))
          & ( Xb = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_6637_find__Some__iff2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),Xb) = find(A,P,Xs) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),I4))
          & ( Xb = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_6638_find__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,P,X3)
            <=> aa(A,$o,Q,X3) ) )
       => ( find(A,P,Xs) = find(A,Q,Ys) ) ) ) ).

% find_cong
tff(fact_6639_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,$o)] : find(A,Uu,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_6640_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( find(A,P,Xs) = none(A) )
    <=> ~ ? [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X2) ) ) ).

% find_None_iff
tff(fact_6641_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( none(A) = find(A,P,Xs) )
    <=> ~ ? [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X2) ) ) ).

% find_None_iff2
tff(fact_6642_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_6643_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_6644_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y222: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_6645_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X21),aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_6646_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_6647_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_6648_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y)),zip(A,B,Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_6649_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_6650_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_6651_nths__singleton,axiom,
    ! [A: $tType,Xb: A,A3: set(nat)] :
      nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)),A3) = $ite(member(nat,zero_zero(nat),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)),nil(A)) ).

% nths_singleton
tff(fact_6652_Cons__listrel1__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: 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)),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),Y),Ys)),listrel1(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2)
          & ( Xs = Ys ) )
        | ( ( Xb = Y )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_6653_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xb: B,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,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,F2,Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),groups4207007520872428315er_sum(B,A,F2,A2,Xs))) ) ).

% horner_sum_simps(2)
tff(fact_6654_lexord__cons__cons,axiom,
    ! [A: $tType,A2: A,Xb: list(A),B2: A,Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),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),Y)),lexord(A,R2))
    <=> ( member(product_prod(A,A),aa(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)),product_Pair(list(A),list(A),Xb),Y),lexord(A,R2)) ) ) ) ).

% lexord_cons_cons
tff(fact_6655_lexord__Nil__left,axiom,
    ! [A: $tType,Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Y),lexord(A,R2))
    <=> ? [A7: A,X2: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),X2) ) ).

% lexord_Nil_left
tff(fact_6656_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_6657_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_6658_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),product_Pair(nat,A,Nb),Xb)),enumerate(A,aa(nat,nat,suc,Nb),Xs)) ).

% enumerate_simps(2)
tff(fact_6659_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_6660_nth__Cons__numeral,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),V2: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_6661_take__Cons__numeral,axiom,
    ! [A: $tType,V2: num,Xb: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),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),V2)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_6662_Cons__in__lex,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: 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)),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),Y),Ys)),lex(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2)
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( Xb = Y )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys),lex(A,R2)) ) ) ) ).

% Cons_in_lex
tff(fact_6663_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_6664_find_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P,Xb),aa(A,option(A),some(A),Xb),find(A,P,Xs)) ).

% find.simps(2)
tff(fact_6665_listrelp_OCons,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,Xb),Y)
     => ( 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),Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_6666_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) ) )
        | ? [X2: A,Y4: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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,X2),Y4)
            & aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs3),Ys4) ) ) ) ).

% listrelp.simps
tff(fact_6667_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,Y3: 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),Y3),Ys3) )
                 => ( aa(B,$o,aa(A,fun(B,$o),R2,X3),Y3)
                   => ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs2),Ys3) ) ) ) ) ) ).

% listrelp.cases
tff(fact_6668_arg__min__list_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A] : arg_min_list(A,B,F2,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_6669_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$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),Y)),shuffles(A,Xs,Ys))),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ).

% Cons_shuffles_subset2
tff(fact_6670_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_6671_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_6672_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Y: A,Ys: list(A)] :
          aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)),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),Y),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Ys))) ) ).

% insort_key.simps(2)
tff(fact_6673_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)),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)),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_6674_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)),product_Pair(list(A),list(A),nil(A)),Ys3)
     => ( ! [Xs2: list(A)] : Xb != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs2),nil(A))
       => ~ ! [X3: A,Xs2: list(A),Y3: A,Ys3: list(A)] : Xb != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)) ) ) ).

% shuffles.cases
tff(fact_6675_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)),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,$o)),list(A)),product_Pair(fun(A,fun(A,$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_6676_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: product_prod(fun(A,B),list(A))] :
          ( ! [F3: fun(A,B),X3: A] : Xb != aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
         => ( ! [F3: fun(A,B),X3: A,Y3: A,Zs2: list(A)] : Xb != aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))
           => ~ ! [A4: fun(A,B)] : Xb != aa(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_6677_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)),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,$o)),list(A)),product_Pair(fun(A,fun(A,$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,$o)),X3: A,Y3: A,Xs2: list(A)] : Xb != aa(list(A),product_prod(fun(A,fun(A,$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),Y3),Xs2))) ) ) ).

% successively.cases
tff(fact_6678_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F3: 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))),product_Pair(fun(A,B),product_prod(list(A),list(B)),F3),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),nil(A)),Bs2))
     => ~ ! [F3: fun(A,B),A4: A,As2: list(A),Bs2: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),product_Pair(fun(A,B),product_prod(list(A),list(B)),F3),aa(list(B),product_prod(list(A),list(B)),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_6679_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_6680_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_6681_list_Oexhaust,axiom,
    ! [A: $tType,Y: list(A)] :
      ( ( Y != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X212),X223) ) ).

% list.exhaust
tff(fact_6682_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_6683_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_6684_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,Y3: 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),Y3),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_6685_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_6686_list__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
     => ( ! [X3: A,Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),nil(B))
       => ( ! [Y3: B,Ys3: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3))
         => ( ! [X3: A,Xs2: list(A),Y3: B,Ys3: list(B)] :
                ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ys3)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,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),Y3),Ys3)) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_6687_list__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),$o)] :
      ( ( Xs != nil(A) )
     => ( ! [X3: A] : aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ( ! [X3: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P,Xs2)
               => aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)) ) )
         => aa(list(A),$o,P,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_6688_removeAll_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Y: 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),Y),Xs)) = $ite(Xb = Y,aa(list(A),list(A),removeAll(A,Xb),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),removeAll(A,Xb),Xs))) ).

% removeAll.simps(2)
tff(fact_6689_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_6690_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_6691_remove1_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A)] :
      remove1(A,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = $ite(Xb = Y,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),remove1(A,Xb,Xs))) ).

% remove1.simps(2)
tff(fact_6692_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_6693_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_6694_remdups__adj_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Y: 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),Y),Xs))) = $ite(Xb = Y,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),Y),Xs)))) ).

% remdups_adj.simps(3)
tff(fact_6695_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),$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),P,nil(A)),nil(B))
       => ( ! [X3: A,Xs2: list(A),Y3: 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),P,Xs2),Ys3)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,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),Y3),Ys3)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_6696_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),$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),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X3: A,Xs2: list(A),Y3: B,Ys3: list(B),Z2: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs2),Ys3),Zs2)
                   => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z2),Zs2)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs),Ys),Zs) ) ) ) ) ).

% list_induct3
tff(fact_6697_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws2: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),$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))),P,nil(A)),nil(B)),nil(C)),nil(D))
           => ( ! [X3: A,Xs2: list(A),Y3: 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))),P,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))),P,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),Y3),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))),P,Xs),Ys),Zs),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_6698_length__Cons,axiom,
    ! [A: $tType,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)),cons(A),Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_6699_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_6700_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_6701_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_6702_insort__key_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),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_6703_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)) ) ) )
           => ~ ! [Y3: A,Ys5: list(A)] :
                  ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
                 => ! [Z2: A,Zs4: list(A)] :
                      ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs4) )
                     => ( ( Y3 = Z2 )
                       => ~ member(list(A),Zs4,shuffles(A,Xs,Ys5)) ) ) ) ) ) ) ) ).

% shufflesE
tff(fact_6704_remdups__adj_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A)] :
      ( ( remdups_adj(A,Xb) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != nil(A) ) )
       => ( ! [X3: A] :
              ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
             => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) ) )
         => ~ ! [X3: A,Y3: 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),Y3),Xs2)) )
               => ( Y != $ite(X3 = Y3,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),Y3),Xs2)))) ) ) ) ) ) ).

% remdups_adj.elims
tff(fact_6705_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_6706_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A)] : shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys))) ).

% shuffles.simps(3)
tff(fact_6707_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)),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)),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_6708_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( member(A,Y,aa(list(A),set(A),set2(A),X222))
     => member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222))) ) ).

% list.set_intros(2)
tff(fact_6709_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_6710_list_Oset__cases,axiom,
    ! [A: $tType,E2: A,A2: list(A)] :
      ( member(A,E2,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),E2),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,E2,aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_6711_set__ConsD,axiom,
    ! [A: $tType,Y: A,Xb: A,Xs: list(A)] :
      ( member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)))
     => ( ( Y = Xb )
        | member(A,Y,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_ConsD
tff(fact_6712_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_6713_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_6714_list__update__code_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ia: nat,Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(nat,nat,suc,Ia),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),list_update(A,Xs,Ia,Y)) ).

% list_update_code(3)
tff(fact_6715_list__update__code_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),zero_zero(nat),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs) ).

% list_update_code(2)
tff(fact_6716_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ia: nat,V2: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ia,V2) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V2),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_acc(A,fun(list(A),fun(A,fun(nat,list(A)))),Xb),Xs),V2),Ia) ).

% list_update.simps(2)
tff(fact_6717_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_6718_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_6719_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) )
           => ! [Y3: B,Ys5: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5) )
               => ( ( Xy = aa(B,product_prod(A,B),product_Pair(A,B,X3),Y3) )
                 => ( Xys != zip(A,B,Xs4,Ys5) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_6720_inj__split__Cons,axiom,
    ! [A: $tType,X5: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_acd(list(A),fun(A,list(A)))),X5) ).

% inj_split_Cons
tff(fact_6721_inj__on__Cons1,axiom,
    ! [A: $tType,Xb: A,A3: set(list(A))] : inj_on(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),A3) ).

% inj_on_Cons1
tff(fact_6722_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_ace(A,fun(list(A),fun(nat,list(A))),Xb),Xs),Nb) ).

% take_Cons
tff(fact_6723_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_6724_tl__Nil,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ? [X2: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) ) ) ).

% tl_Nil
tff(fact_6725_Nil__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),tl(A),Xs) )
    <=> ( ( Xs = nil(A) )
        | ? [X2: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) ) ) ).

% Nil_tl
tff(fact_6726_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_6727_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ys: list(A),Xs: list(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),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_6728_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_6729_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_acf(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_6730_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Y: A,Zs: list(A)] :
          arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = $let(
            m2: A,
            m2:= arg_min_list(A,B,F2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,m2)),Xb,m2) ) ) ).

% arg_min_list.simps(2)
tff(fact_6731_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X2: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys4) )
          & aa(nat,$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_6732_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B),A2: A] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),aa(A,B,F2,X3)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),A2),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_6733_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: 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)),product_Pair(list(A),list(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R2))
     => ( ! [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),product_Pair(A,A,X3),Y),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Zs2),Ys),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_6734_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)),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))
     => ( ! [Y3: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Xs) )
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y3),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)),product_Pair(list(A),list(A),Xs),Zs2),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_6735_listrel1I1,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2)
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)),listrel1(A,R2)) ) ).

% listrel1I1
tff(fact_6736_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: 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)),product_Pair(list(A),list(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R2))
     => ~ ! [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),product_Pair(A,B,X3),Y),R2)
             => ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs2),Ys),listrel(A,B,R2)) ) ) ) ).

% listrel_Cons2
tff(fact_6737_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: 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)),product_Pair(list(A),list(B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xs),listrel(A,B,R2))
     => ~ ! [Y3: B,Ys3: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys3) )
           => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Y),Y3),R2)
             => ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Ys),Ys3),listrel(A,B,R2)) ) ) ) ).

% listrel_Cons1
tff(fact_6738_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,R2: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xb),Y),R2)
     => ( member(product_prod(list(A),list(B)),aa(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)),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),Y),Ys)),listrel(A,B,R2)) ) ) ).

% listrel.Cons
tff(fact_6739_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_6740_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Y) = $ite(Xb = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)),aa(A,nat,count_list(A,Xs),Y)) ).

% count_list.simps(2)
tff(fact_6741_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_6742_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_6743_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_6744_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)),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)),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_6745_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_acg(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_6746_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_6747_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_6748_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)),product_Pair(list(A),list(B),A1),A22),listrel(A,B,R2))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X3: A,Y3: 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),Y3),Ys3) )
                 => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y3),R2)
                   => ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs2),Ys3),listrel(A,B,R2)) ) ) ) ) ) ).

% listrel.cases
tff(fact_6749_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)),product_Pair(list(A),list(B),A1),A22),listrel(A,B,R2))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X2: A,Y4: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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),product_Pair(A,B,X2),Y4),R2)
            & member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs3),Ys4),listrel(A,B,R2)) ) ) ) ).

% listrel.simps
tff(fact_6750_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_6751_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_6752_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_6753_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_6754_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_6755_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Ia)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,Ia,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,Ia)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,Ia),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_6756_shuffles_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,Xb,Xaa) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xaa),bot_bot(set(list(A)))) ) )
       => ( ( ( Xaa = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(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) )
               => ! [Y3: A,Ys3: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                   => ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys3))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_6757_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Ia)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,Ia,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,Ia)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,Ia),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_6758_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_6759_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A),Nb: nat] :
      ( ( Xb != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6760_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_6761_Cons__replicate__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = replicate(A,Nb,Y) )
    <=> ( ( Xb = Y )
        & aa(nat,$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_6762_Cons__lenlex__iff,axiom,
    ! [A: $tType,Mb: A,Ms: list(A),Nb: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Mb),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Nb),Ns)),lenlex(A,R2))
    <=> ( aa(nat,$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),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)),product_Pair(list(A),list(A),Ms),Ns),lenlex(A,R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6763_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_6764_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),transitive_rtrancl(A,R2))
     => ( member(product_prod(list(A),list(A)),aa(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)),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),Y),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_6765_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(
        a2: set(set(A)),
        a2:= pow2(A,aa(list(A),set(A),set2(A),Xs)),
        aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a2),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb)),a2)) ) ).

% Pow_set(2)
tff(fact_6766_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set(A)] : set_Cons(A,A3,aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_ach(A,list(A))),A3) ).

% set_Cons_sing_Nil
tff(fact_6767_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_6768_append_Oassoc,axiom,
    ! [A: $tType,A2: list(A),B2: list(A),C2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),B2)),C2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),B2),C2)) ).

% append.assoc
tff(fact_6769_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_6770_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_6771_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_6772_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_6773_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_6774_self__append__conv2,axiom,
    ! [A: $tType,Y: list(A),Xs: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Y) )
    <=> ( Xs = nil(A) ) ) ).

% self_append_conv2
tff(fact_6775_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_6776_self__append__conv,axiom,
    ! [A: $tType,Y: list(A),Ys: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Y),Ys) )
    <=> ( Ys = nil(A) ) ) ).

% self_append_conv
tff(fact_6777_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_6778_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_6779_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_6780_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_6781_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_6782_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_6783_append1__eq__conv,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),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),Y),nil(A))) )
    <=> ( ( Xs = Ys )
        & ( Xb = Y ) ) ) ).

% append1_eq_conv
tff(fact_6784_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_6785_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_6786_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_6787_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_6788_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_6789_fun__upds__append__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Mb: fun(A,option(B)),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,Mb,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs),Ys) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_6790_fun__upds__append2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Mb: fun(A,option(B)),Zs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,Mb,Xs,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_6791_size__list__append,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_list(A,F2),Xs)),aa(list(A),nat,size_list(A,F2),Ys)) ).

% size_list_append
tff(fact_6792_bind__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xb: B,Xs: list(B),F2: fun(B,list(A))] : bind(B,A,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs),F2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F2,Xb)),bind(B,A,Xs,F2)) ).

% bind_simps(2)
tff(fact_6793_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_6794_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_6795_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_6796_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: 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),Y) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ).

% list_update_length
tff(fact_6797_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_6798_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_6799_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_6800_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_6801_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),Xs5: 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),Xs5)),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),Xs5),concat(A,Xss22)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_6802_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_6803_rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [X3: A,Xs2: list(A)] :
            ( aa(list(A),$o,P,Xs2)
           => aa(list(A),$o,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))) )
       => aa(list(A),$o,P,Xs) ) ) ).

% rev_induct
tff(fact_6804_rev__exhaust,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ~ ! [Ys3: list(A),Y3: 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),Y3),nil(A))) ) ).

% rev_exhaust
tff(fact_6805_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 ) )
        | ? [Ys6: list(A)] :
            ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys6) = Ys )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_6806_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) ) )
        | ? [Ys6: list(A)] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys6) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_6807_rev__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),$o)] :
      ( ( Xs != nil(A) )
     => ( ! [X3: A] : aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ( ! [X3: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P,Xs2)
               => aa(list(A),$o,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))) ) )
         => aa(list(A),$o,P,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_6808_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_6809_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_6810_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_6811_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X2) )
    <=> ? [Ys4: list(A),X2: A] :
          ( ? [Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs3))
          & aa(A,$o,P,X2)
          & ! [Xa3: A] :
              ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys4))
             => ~ aa(A,$o,P,Xa3) ) ) ) ).

% split_list_first_prop_iff
tff(fact_6812_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X2) )
    <=> ? [Ys4: list(A),X2: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs3)) )
          & aa(A,$o,P,X2)
          & ! [Xa3: A] :
              ( member(A,Xa3,aa(list(A),set(A),set2(A),Zs3))
             => ~ aa(A,$o,P,Xa3) ) ) ) ).

% split_list_last_prop_iff
tff(fact_6813_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_6814_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_6815_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ~ ! [Ys3: list(A),X3: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
           => ( aa(A,$o,P,X3)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_first_propE
tff(fact_6816_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ~ ! [Ys3: list(A),X3: A,Zs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2)) )
           => ( aa(A,$o,P,X3)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_last_propE
tff(fact_6817_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ? [Ys3: list(A),X3: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
          & aa(A,$o,P,X3)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_first_prop
tff(fact_6818_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ? [Ys3: list(A),X3: A,Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2)) )
          & aa(A,$o,P,X3)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_last_prop
tff(fact_6819_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_6820_append__Cons__eq__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),Xs6: list(A),Ys7: 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),Xs6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys7)) )
        <=> ( ( Xs = Xs6 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_6821_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ~ ! [Ys3: list(A),X3: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
           => ~ aa(A,$o,P,X3) ) ) ).

% split_list_propE
tff(fact_6822_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_6823_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ? [Ys3: list(A),X3: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
          & aa(A,$o,P,X3) ) ) ).

% split_list_prop
tff(fact_6824_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_6825_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_6826_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_6827_longest__common__prefix,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
    ? [Ps: list(A),Xs4: list(A),Ys5: list(A)] :
      ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps),Xs4) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps),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_6828_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_6829_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_6830_append__replicate__commute,axiom,
    ! [A: $tType,Nb: nat,Xb: A,K: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),replicate(A,K,Xb)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,K,Xb)),replicate(A,Nb,Xb)) ).

% append_replicate_commute
tff(fact_6831_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_6832_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_6833_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V2: 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)),product_Pair(list(A),list(A),U),V2),lexord(A,R2))
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(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),V2)),lexord(A,R2)) ) ).

% lexord_append_leftI
tff(fact_6834_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)),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)),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)),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_6835_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_6836_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_6837_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_6838_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_6839_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_6840_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_6841_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_6842_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)),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)),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_6843_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) )
     => ? [M2: nat,N: nat,Zs2: list(A)] :
          ( ( concat(A,replicate(list(A),M2,Zs2)) = Xs )
          & ( concat(A,replicate(list(A),N,Zs2)) = Ys ) ) ) ).

% comm_append_are_replicate
tff(fact_6844_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),Y3: A,Ys5: list(A)] :
            ( ( X3 != Y3 )
            & ( 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),Y3),nil(A))),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_6845_not__distinct__decomp,axiom,
    ! [A: $tType,Ws2: list(A)] :
      ( ~ distinct(A,Ws2)
     => ? [Xs2: list(A),Ys3: list(A),Zs2: list(A),Y3: 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),Y3),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),Y3),nil(A))),Zs2)))) ) ).

% not_distinct_decomp
tff(fact_6846_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_6847_replicate__append__same,axiom,
    ! [A: $tType,Ia: nat,Xb: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Ia,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,Ia,Xb)) ).

% replicate_append_same
tff(fact_6848_list__update__append1,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),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),Ia,Xb) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,Ia,Xb)),Ys) ) ) ).

% list_update_append1
tff(fact_6849_remdups__adj__append__two,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Y: A] :
      remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))))),
        $ite(Xb = Y,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ).

% remdups_adj_append_two
tff(fact_6850_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_6851_lexord__append__leftD,axiom,
    ! [A: $tType,Xb: list(A),U: list(A),V2: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),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),V2)),lexord(A,R2))
     => ( ! [A4: A] : ~ member(product_prod(A,A),aa(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)),product_Pair(list(A),list(A),U),V2),lexord(A,R2)) ) ) ).

% lexord_append_leftD
tff(fact_6852_lexord__append__rightI,axiom,
    ! [A: $tType,Y: list(A),Xb: list(A),R2: set(product_prod(A,A))] :
      ( ? [B10: A,Z3: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B10),Z3)
     => member(product_prod(list(A),list(A)),aa(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),Y)),lexord(A,R2)) ) ).

% lexord_append_rightI
tff(fact_6853_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)),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)),product_Pair(list(A),list(A),Xs),Ys),lexord(A,R2)) ) ) ) ) ).

% lexord_sufE
tff(fact_6854_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),product_Pair(A,A,X3),X3),R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(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)),product_Pair(list(A),list(A),Ys),Zs),lex(A,R2)) ) ) ).

% lex_append_left_iff
tff(fact_6855_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),product_Pair(A,A,X3),X3),R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(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)),product_Pair(list(A),list(A),Ys),Zs),lex(A,R2)) ) ) ).

% lex_append_leftD
tff(fact_6856_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_6857_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)),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)),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_6858_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)),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)),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_6859_nths__append,axiom,
    ! [A: $tType,L: list(A),L2: list(A),A3: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L2),A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A3)),nths(A,L2,collect(nat,aa(set(nat),fun(nat,$o),aTP_Lamp_aci(list(A),fun(set(nat),fun(nat,$o)),L),A3)))) ).

% nths_append
tff(fact_6860_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_6861_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_6862_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_6863_listset_Osimps_I2_J,axiom,
    ! [A: $tType,A3: set(A),As3: list(set(A))] : listset(A,aa(list(set(A)),list(set(A)),aa(set(A),fun(list(set(A)),list(set(A))),cons(set(A)),A3),As3)) = set_Cons(A,A3,listset(A,As3)) ).

% listset.simps(2)
tff(fact_6864_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_6865_listrel1I,axiom,
    ! [A: $tType,Xb: A,Y: 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),product_Pair(A,A,Xb),Y),R2)
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),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),Y),Vs)) )
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys),listrel1(A,R2)) ) ) ) ).

% listrel1I
tff(fact_6866_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)),product_Pair(list(A),list(A),Xs),Ys),listrel1(A,R2))
     => ~ ! [X3: A,Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y3),R2)
           => ! [Us3: list(A),Vs2: 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),Vs2)) )
               => ( 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),Y3),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_6867_lexord__append__left__rightI,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),U: list(A),Xb: list(A),Y: list(A)] :
      ( member(product_prod(A,A),aa(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)),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),Y))),lexord(A,R2)) ) ).

% lexord_append_left_rightI
tff(fact_6868_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)),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))
    <=> ( ? [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X2),X2),R2) )
        | member(product_prod(list(A),list(A)),aa(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_6869_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R2: set(product_prod(A,A)),V2: list(A),Z: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),U),W2),lexord(A,R2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),Z)),lexord(A,R2)) ) ) ).

% lexord_sufI
tff(fact_6870_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_6871_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),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),Y),nil(A)))),listrel1(A,R2))
    <=> ( ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys),listrel1(A,R2))
          & ( Xb = Y ) )
        | ( ( Xs = Ys )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_6872_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups4207007520872428315er_sum(B,A,F2,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,F2,A2,Ys))) ) ).

% horner_sum_append
tff(fact_6873_nths__Cons,axiom,
    ! [A: $tType,Xb: A,L: list(A),A3: set(nat)] :
      nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),L),A3) = aa(list(A),list(A),
        aa(list(A),fun(list(A),list(A)),append(A),
          $ite(member(nat,zero_zero(nat),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)),nil(A))),
        nths(A,L,collect(nat,aTP_Lamp_acj(set(nat),fun(nat,$o),A3)))) ).

% nths_Cons
tff(fact_6874_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_6875_set__Cons__def,axiom,
    ! [A: $tType,A3: set(A),XS: set(list(A))] : set_Cons(A,A3,XS) = collect(list(A),aa(set(list(A)),fun(list(A),$o),aTP_Lamp_ack(set(A),fun(set(list(A)),fun(list(A),$o)),A3),XS)) ).

% set_Cons_def
tff(fact_6876_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
              & ( concat(A,replicate(list(A),N,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_6877_listrel1__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_acl(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% listrel1_def
tff(fact_6878_lexord__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_acm(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lexord_def
tff(fact_6879_lex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_acn(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lex_conv
tff(fact_6880_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,aa(nat,nat,suc,Ia),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Ia,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),Ia)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_6881_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_6882_pos__n__replace,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Y: A] :
      ( aa(nat,$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),Y),nil(A))),drop(A,aa(nat,nat,suc,Nb),Xs)))) ) ) ).

% pos_n_replace
tff(fact_6883_drop0,axiom,
    ! [A: $tType,X: list(A)] : drop(A,zero_zero(nat),X) = X ).

% drop0
tff(fact_6884_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_6885_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_6886_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_6887_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_6888_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_6889_drop__replicate,axiom,
    ! [A: $tType,Ia: nat,K: nat,Xb: A] : drop(A,Ia,replicate(A,K,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),Ia),Xb) ).

% drop_replicate
tff(fact_6890_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_6891_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_6892_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_6893_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_6894_drop__Cons__numeral,axiom,
    ! [A: $tType,V2: num,Xb: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_6895_nth__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ia: nat] :
      ( aa(nat,$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)),Ia) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Ia)) ) ) ).

% nth_drop
tff(fact_6896_nth__via__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Y: A,Ys: list(A)] :
      ( ( drop(A,Nb,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
     => ( aa(nat,A,nth(A,Xs),Nb) = Y ) ) ).

% nth_via_drop
tff(fact_6897_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_6898_drop__eq__nths,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,Nb,Xs) = nths(A,Xs,collect(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb))) ).

% drop_eq_nths
tff(fact_6899_drop__Nil,axiom,
    ! [A: $tType,Nb: nat] : drop(A,Nb,nil(A)) = nil(A) ).

% drop_Nil
tff(fact_6900_distinct__drop,axiom,
    ! [A: $tType,Xs: list(A),Ia: nat] :
      ( distinct(A,Xs)
     => distinct(A,drop(A,Ia,Xs)) ) ).

% distinct_drop
tff(fact_6901_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_6902_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_6903_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_6904_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_6905_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_6906_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_6907_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_6908_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_6909_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_6910_take__add,axiom,
    ! [A: $tType,Ia: nat,J: nat,Xs: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),J),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Ia,Xs)),take(A,J,drop(A,Ia,Xs))) ).

% take_add
tff(fact_6911_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_6912_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_aco(list(A),fun(nat,list(A)),Xs),Nb) ).

% drop_Cons
tff(fact_6913_nths__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I5: set(nat)] : nths(A,drop(A,Nb,Xs),I5) = nths(A,Xs,aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb)),I5)) ).

% nths_drop
tff(fact_6914_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_6915_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_6916_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_6917_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_6918_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_6919_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),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),Ia)),drop(A,aa(nat,nat,suc,Ia),Xs)) = drop(A,Ia,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_6920_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_6921_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),Ia: nat,J: nat] :
      ( distinct(A,Vs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),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,Ia,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_6922_id__take__nth__drop,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),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,Ia,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),Ia)),drop(A,aa(nat,nat,suc,Ia),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_6923_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A),A2: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,Xs,Ia,A2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Ia,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),drop(A,aa(nat,nat,suc,Ia),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_6924_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_6925_lexn__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Nb: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),Nb) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_acp(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),Nb))) ).

% lexn_conv
tff(fact_6926_list__encode_Oelims,axiom,
    ! [Xb: list(nat),Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( ( ( Xb = nil(nat) )
         => ( Y != 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) )
             => ( Y != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_6927_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_6928_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_6929_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)),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_6930_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_6931_upto__aux__rec,axiom,
    ! [Ia: int,J: int,Js: list(int)] :
      upto_aux(Ia,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia),Js,upto_aux(Ia,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_6932_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),Ys),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Y),Zs))) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & aa(A,$o,P,Y)
        & ~ ? [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X2) ) ) ) ).

% extract_Some_iff
tff(fact_6933_extract__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( extract(A,P,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X2) ) ) ).

% extract_None_iff
tff(fact_6934_extract__Nil__code,axiom,
    ! [A: $tType,P: fun(A,$o)] : extract(A,P,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ).

% extract_Nil_code
tff(fact_6935_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P,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))),product_Pair(list(A),product_prod(A,list(A)),nil(A)),aa(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)))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))),aTP_Lamp_acr(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Xb)),extract(A,P,Xs))) ).

% extract_Cons_code
tff(fact_6936_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),Ys),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Y),Zs))) )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & aa(A,$o,P,Y)
        & ~ ? [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X) ) ) ) ).

% extract_SomeE
tff(fact_6937_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,Xb),Xaa))
       => ~ ( ( Y = $ite(aa(int,$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),product_Pair(int,int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_6938_upto_Opsimps,axiom,
    ! [Ia: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,Ia),J))
     => ( upto(Ia,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Ia),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_6939_upto__empty,axiom,
    ! [J: int,Ia: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia)
     => ( upto(Ia,J) = nil(int) ) ) ).

% upto_empty
tff(fact_6940_upto__Nil2,axiom,
    ! [Ia: int,J: int] :
      ( ( nil(int) = upto(Ia,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia) ) ).

% upto_Nil2
tff(fact_6941_upto__Nil,axiom,
    ! [Ia: int,J: int] :
      ( ( upto(Ia,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),Ia) ) ).

% upto_Nil
tff(fact_6942_upto__single,axiom,
    ! [Ia: int] : upto(Ia,Ia) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Ia),nil(int)) ).

% upto_single
tff(fact_6943_nth__upto,axiom,
    ! [Ia: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(Ia,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_6944_length__upto,axiom,
    ! [Ia: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(Ia,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),Ia)),one_one(int))) ).

% length_upto
tff(fact_6945_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_6946_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_6947_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_6948_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_6949_upto__code,axiom,
    ! [Ia: int,J: int] : upto(Ia,J) = upto_aux(Ia,J,nil(int)) ).

% upto_code
tff(fact_6950_upto__aux__def,axiom,
    ! [Ia: int,J: int,Js: list(int)] : upto_aux(Ia,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,J)),Js) ).

% upto_aux_def
tff(fact_6951_atLeastAtMost__upto,axiom,
    ! [Ia: int,J: int] : set_or1337092689740270186AtMost(int,Ia,J) = aa(list(int),set(int),set2(int),upto(Ia,J)) ).

% atLeastAtMost_upto
tff(fact_6952_distinct__upto,axiom,
    ! [Ia: int,J: int] : distinct(int,upto(Ia,J)) ).

% distinct_upto
tff(fact_6953_upto__split2,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(Ia,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_6954_upto__split1,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(Ia,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_6955_atLeastLessThan__upto,axiom,
    ! [Ia: int,J: int] : set_or7035219750837199246ssThan(int,Ia,J) = aa(list(int),set(int),set2(int),upto(Ia,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6956_greaterThanAtMost__upto,axiom,
    ! [Ia: int,J: int] : set_or3652927894154168847AtMost(int,Ia,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_6957_upto__rec1,axiom,
    ! [Ia: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( upto(Ia,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Ia),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6958_upto_Osimps,axiom,
    ! [Ia: int,J: int] :
      upto(Ia,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Ia),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_6959_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa(int,$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_6960_upto__rec2,axiom,
    ! [Ia: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( upto(Ia,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,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_6961_greaterThanLessThan__upto,axiom,
    ! [Ia: int,J: int] : set_or5935395276787703475ssThan(int,Ia,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_6962_upto__split3,axiom,
    ! [Ia: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ia),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(Ia,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(Ia,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_6963_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(A),list(product_prod(A,B)),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_acs(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).

% zip_Cons
tff(fact_6964_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_act(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Xb),Xs)),Ys) ).

% zip_Cons1
tff(fact_6965_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_6966_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_acu(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),H),F22)),List) ).

% list.case_distrib
tff(fact_6967_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_6968_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_acv(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_6969_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_acw(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_6970_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_acx(A,fun(A,fun(list(A),list(A))),Xb)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_6971_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_6972_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_acy(A,fun(list(A),fun(A,fun(list(A),A))),Xb),Xs)),Xs) ) ).

% min_list.simps
tff(fact_6973_list_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,P,aa(list(B),A,case_list(A,B,F1,F22),List))
    <=> ( ( ( List = nil(B) )
         => aa(A,$o,P,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,P,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_6974_list_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,P,aa(list(B),A,case_list(A,B,F1,F22),List))
    <=> ~ ( ( ( List = nil(B) )
            & ~ aa(A,$o,P,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,P,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_6975_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_6976_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)),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)),product_Pair(list(A),list(A),Xs),Ys),lenlex(A,R)) ) ) ).

% lenlex_append2
tff(fact_6977_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)),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)),product_Pair(list(A),list(A),Ys),Zs),lexord(A,R2)) ) ) ).

% lexord_same_pref_if_irrefl
tff(fact_6978_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_acz(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_6979_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_ada(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_6980_lexord__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
     => irrefl(list(A),lexord(A,R)) ) ).

% lexord_irrefl
tff(fact_6981_irrefl__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
     => irrefl(list(A),lex(A,R2)) ) ).

% irrefl_lex
tff(fact_6982_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)),product_Pair(list(A),list(A),Xb),Xb),lex(A,R2)) ) ).

% lexl_not_refl
tff(fact_6983_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_6984_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_adb(nat,fun(nat,complex),Nb),aa(nat,set(nat),set_ord_lessThan(nat),Nb),collect(complex,aTP_Lamp_cs(nat,fun(complex,$o),Nb))) ) ).

% bij_betw_roots_unity
tff(fact_6985_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != 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,C2)))),cis(divide_divide(real,arg(C2),aa(nat,real,semiring_1_of_nat(real),Nb))))),collect(complex,aTP_Lamp_cs(nat,fun(complex,$o),Nb)),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_jp(complex,fun(nat,fun(complex,$o)),C2),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_6986_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A),B3: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3,B3)
        <=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A2)),A3) = B3 ) ) ) ).

% bij_betw_add
tff(fact_6987_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N3: set(nat),A3: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N3,A3)
        <=> ( aa(set(nat),set(A),image2(nat,A,semiring_1_of_nat(A)),N3) = A3 ) ) ) ).

% bij_betw_of_nat
tff(fact_6988_bij__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_6989_bij__betw__funpow,axiom,
    ! [A: $tType,F2: fun(A,A),S2: set(A),Nb: nat] :
      ( bij_betw(A,A,F2,S2,S2)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),S2,S2) ) ).

% bij_betw_funpow
tff(fact_6990_bij__betw__finite,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( bij_betw(A,B,F2,A3,B3)
     => ( aa(set(A),$o,finite_finite2(A),A3)
      <=> aa(set(B),$o,finite_finite2(B),B3) ) ) ).

% bij_betw_finite
tff(fact_6991_finite__same__card__bij,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) )
         => ? [H4: fun(A,B)] : bij_betw(A,B,H4,A3,B3) ) ) ) ).

% finite_same_card_bij
tff(fact_6992_bij__betw__iff__card,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( ? [F7: fun(A,B)] : bij_betw(A,B,F7,A3,B3)
        <=> ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) ) ) ) ) ).

% bij_betw_iff_card
tff(fact_6993_bij__betw__same__card,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B)] :
      ( bij_betw(A,B,F2,A3,B3)
     => ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) ) ) ).

% bij_betw_same_card
tff(fact_6994_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F9: fun(A,B),A11: set(A),A15: set(B),F2: fun(C,A),A3: set(C)] :
      ( bij_betw(A,B,F9,A11,A15)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),A3)),A11)
       => ( bij_betw(C,A,F2,A3,A11)
        <=> bij_betw(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,F9),F2),A3,A15) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_6995_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A3: set(A),F2: fun(A,B),A11: set(B)] :
      ( ~ member(A,B2,A3)
     => ( ~ member(B,aa(A,B,F2,B2),A11)
       => ( bij_betw(A,B,F2,A3,A11)
        <=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A11),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_6996_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A3: set(A),F2: fun(A,B),A11: set(B)] :
      ( ~ member(A,B2,A3)
     => ( ~ member(B,aa(A,B,F2,B2),A11)
       => ( bij_betw(A,B,F2,A3,A11)
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A11),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_6997_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C5: set(A),B3: set(B),D5: set(B)] :
      ( bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D5))
     => ( bij_betw(A,B,F2,C5,D5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D5) = bot_bot(set(B)) )
           => bij_betw(A,B,F2,A3,B3) ) ) ) ) ).

% bij_betw_partition
tff(fact_6998_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B),C5: set(A),D5: set(B)] :
      ( bij_betw(A,B,F2,A3,B3)
     => ( bij_betw(A,B,F2,C5,D5)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B3),D5) = bot_bot(set(B)) )
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),C5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B3),D5)) ) ) ) ).

% bij_betw_combine
tff(fact_6999_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] :
      ( bij_betw(A,B,F2,A3,bot_bot(set(B)))
     => ( A3 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_7000_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( bij_betw(A,B,F2,bot_bot(set(A)),A3)
     => ( A3 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_7001_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),A11: set(B),B3: set(A),B14: set(B)] :
      ( bij_betw(A,B,F2,A3,A11)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
       => ( ( aa(set(A),set(B),image2(A,B,F2),B3) = B14 )
         => bij_betw(A,B,F2,B3,B14) ) ) ) ).

% bij_betw_subset
tff(fact_7002_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F9: fun(B,A),F2: fun(A,B),A11: set(B)] :
      ( ! [X3: A] :
          ( member(A,X3,A3)
         => ( aa(B,A,F9,aa(A,B,F2,X3)) = X3 ) )
     => ( ! [X3: B] :
            ( member(B,X3,A11)
           => ( aa(A,B,F2,aa(B,A,F9,X3)) = X3 ) )
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),A11)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F9),A11)),A3)
           => bij_betw(A,B,F2,A3,A11) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_7003_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C5: set(B),G: fun(A,B),B3: set(A),D5: set(B)] :
      ( bij_betw(A,B,F2,A3,C5)
     => ( bij_betw(A,B,G,B3,D5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),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_zz(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C5),D5)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_7004_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F2: fun(B,C),A11: fun(A,set(C))] :
      ( ! [I2: A,J2: A] :
          ( member(A,I2,I5)
         => ( member(A,J2,I5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,I2)),aa(A,set(B),A3,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I2)) ) ) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => bij_betw(B,C,F2,aa(A,set(B),A3,I2),aa(A,set(C),A11,I2)) )
       => bij_betw(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A3),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A11),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_7005_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ? [H4: fun(A,A)] : bij_betw(A,A,H4,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_7006_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ? [H4: fun(A,A)] : bij_betw(A,A,H4,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_7007_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_7008_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_7009_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T4: set(B),H: fun(A,B),S2: set(A),T3: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S4)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T4))
             => ( ! [A4: A] :
                    ( member(A,A4,S4)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = zero_zero(C) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T4)
                     => ( aa(B,C,G,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_adc(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),T3) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_7010_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T4: set(B),H: fun(A,B),S2: set(A),T3: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S4)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T4))
             => ( ! [A4: A] :
                    ( member(A,A4,S4)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = one_one(C) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T4)
                     => ( aa(B,C,G,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_add(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T3) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_7011_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B3: set(A)] :
      ( distinct(A,Xs)
     => ( ( A3 = aa(nat,set(nat),set_ord_lessThan(nat),aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B3 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A3,B3) ) ) ) ).

% bij_betw_nth
tff(fact_7012_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_7013_sum_OatLeastAtMost__reindex,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add(B)
        & ord(A) )
     => ! [H: fun(nat,A),Mb: nat,Nb: nat,G: 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),G),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,G),H)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ) ) ).

% sum.atLeastAtMost_reindex
tff(fact_7014_sum_OatLeastLessThan__reindex,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_add(B)
        & ord(A) )
     => ! [H: fun(nat,A),Mb: nat,Nb: nat,G: 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),G),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,G),H)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ) ) ).

% sum.atLeastLessThan_reindex
tff(fact_7015_prod_OatLeastAtMost__reindex,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ord(A) )
     => ! [H: fun(nat,A),Mb: nat,Nb: nat,G: 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),G),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,G),H)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ) ) ).

% prod.atLeastAtMost_reindex
tff(fact_7016_prod_OatLeastLessThan__reindex,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ord(A) )
     => ! [H: fun(nat,A),Mb: nat,Nb: nat,G: 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),G),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,G),H)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ) ) ).

% prod.atLeastLessThan_reindex
tff(fact_7017_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B3: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3)
       => ( inj_on(B,A,G,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),B3)),A3)
           => ? [H4: fun(A,B)] : bij_betw(A,B,H4,A3,B3) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_7018_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_ade(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_7019_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_7020_some__in__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( member(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,$o),A3)),A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_7021_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A3) )
     => ( ? [X_13: A] : aa(A,$o,P,X_13)
      <=> (A3) ) ) ).

% verit_sko_ex'
tff(fact_7022_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_13: A] : aa(A,$o,P,X_13)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_be(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_7023_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_be(fun(A,$o),fun(A,$o),P)))
      <=> (A3) )
     => ( ! [X_13: A] : aa(A,$o,P,X_13)
      <=> (A3) ) ) ).

% verit_sko_forall'
tff(fact_7024_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B3: A,A3: A,P: fun(A,$o)] :
      ( ( B3 = A3 )
     => ( ( fChoice(A,P) = A3 )
      <=> ( fChoice(A,P) = B3 ) ) ) ).

% verit_sko_forall''
tff(fact_7025_verit__sko__ex__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ? [X_13: A] : aa(A,$o,P,X_13)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_ex_indirect
tff(fact_7026_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P2: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ! [X3: A] :
            ( aa(A,$o,P,X3)
          <=> aa(A,$o,P2,X3) )
       => ( ? [X_13: A] : aa(A,$o,P2,X_13)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_7027_verit__sko__forall__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_be(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_13: A] : aa(A,$o,P,X_13)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_forall_indirect
tff(fact_7028_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P2: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_be(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X3: A] :
            ( aa(A,$o,P,X3)
          <=> aa(A,$o,P2,X3) )
       => ( ! [X_13: A] : aa(A,$o,P2,X_13)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_7029_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_7030_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_7031_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( lattic7623131987881927897min_on(A,B,F2,S2) = fChoice(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_adf(set(A),fun(fun(A,B),fun(A,$o)),S2),F2)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_7032_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Y: A] :
          ( ( min_list(A,Xb) = Y )
         => ( ! [X3: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( Y != aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_acy(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) ) )
           => ~ ( ( Xb = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_7033_lists__length__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] : collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_adg(set(A),fun(nat,fun(list(A),$o)),A3),Nb)) = aa(set(product_prod(list(A),A)),set(list(A)),image2(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_acd(list(A),fun(A,list(A))))),product_Sigma(list(A),A,collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_at(set(A),fun(nat,fun(list(A),$o)),A3),Nb)),aTP_Lamp_adh(set(A),fun(list(A),set(A)),A3))) ).

% lists_length_Suc_eq
tff(fact_7034_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B3: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B3) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_7035_Times__empty,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( ( product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( B3 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_7036_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A3: set(A)] : product_Sigma(A,B,A3,aTP_Lamp_adj(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_7037_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [A4: A] :
            ( member(A,A4,A3)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,A4)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,B3)) ) ) ).

% finite_SigmaI
tff(fact_7038_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_adk(list(B),fun(A,set(B)),Ys)) ).

% set_product
tff(fact_7039_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ! [X3: A] :
            ( member(A,X3,A3)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,X3)) )
       => ( aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A3,B3)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_na(fun(A,set(B)),fun(A,nat),B3)),A3) ) ) ) ).

% card_SigmaI
tff(fact_7040_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,Xb: A,C5: set(A),A3: set(B),B3: set(B)] :
      ( member(A,Xb,C5)
     => ( aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A3,aTP_Lamp_mn(set(A),fun(B,set(A)),C5))),product_Sigma(B,A,B3,aTP_Lamp_mn(set(A),fun(B,set(A)),C5)))
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),B3) ) ) ).

% Times_subset_cancel2
tff(fact_7041_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C5: set(A),B3: fun(A,set(B)),D5: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
     => ( ! [X3: A] :
            ( member(A,X3,A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B3,X3)),aa(A,set(B),D5,X3)) )
       => aa(set(product_prod(A,B)),$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,A3,B3)),product_Sigma(A,B,C5,D5)) ) ) ).

% Sigma_mono
tff(fact_7042_Restr__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B3,aTP_Lamp_adl(set(A),fun(A,set(A)),B3)))),product_Sigma(A,A,A3,aTP_Lamp_adl(set(A),fun(A,set(A)),A3))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_adl(set(A),fun(A,set(A)),A3))) ) ) ).

% Restr_subset
tff(fact_7043_finite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3))) ) ) ).

% finite_cartesian_product
tff(fact_7044_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I5: set(A),X5: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I5,X5) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X2: A] :
          ( member(A,X2,I5)
         => ( aa(A,set(B),X5,X2) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_7045_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),C5: set(A),D5: set(B)] :
      ( ( product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)) = product_Sigma(A,B,C5,aTP_Lamp_adi(set(B),fun(A,set(B)),D5)) )
    <=> ( ( ( A3 = C5 )
          & ( B3 = D5 ) )
        | ( ( ( A3 = bot_bot(set(A)) )
            | ( B3 = bot_bot(set(B)) ) )
          & ( ( C5 = bot_bot(set(A)) )
            | ( D5 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_7046_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),C5: set(B),B3: set(A),D5: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B3,aTP_Lamp_adi(set(B),fun(A,set(B)),D5)))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( C5 = bot_bot(set(B)) )
        | ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C5),D5) ) ) ) ).

% times_subset_iff
tff(fact_7047_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_adm(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_7048_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_adn(set(A),fun(fun(A,set(B)),fun(A,$o)),A3),B3)))
     => ( ! [A4: A] :
            ( member(A,A4,A3)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,A4)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,B3)) ) ) ).

% finite_SigmaI2
tff(fact_7049_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)))
     => ( ( B3 != bot_bot(set(B)) )
       => aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% finite_cartesian_productD1
tff(fact_7050_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)))
     => ( ( A3 != bot_bot(set(A)) )
       => aa(set(B),$o,finite_finite2(B),B3) ) ) ).

% finite_cartesian_productD2
tff(fact_7051_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( B3 = bot_bot(set(B)) )
        | ( aa(set(A),$o,finite_finite2(A),A3)
          & aa(set(B),$o,finite_finite2(B),B3) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_7052_Sigma__interval__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [A3: set(A),V2: 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,A3,aTP_Lamp_ado(fun(A,B),fun(A,set(B)),V2))),product_Sigma(A,B,A3,aa(B,fun(A,set(B)),aTP_Lamp_adp(fun(A,B),fun(B,fun(A,set(B))),V2),W2))) = bot_bot(set(product_prod(A,B))) ) ).

% Sigma_interval_disjoint
tff(fact_7053_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,Xb: A,A3: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))),aTP_Lamp_adi(set(B),fun(A,set(B)),A3))) = aa(set(B),nat,finite_card(B),A3) ).

% card_cartesian_product_singleton
tff(fact_7054_sum_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B3: fun(A,set(B)),G: fun(A,fun(B,C))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,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_adq(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B3),G)),A3) = 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),product_case_prod(A,B,C,G)),product_Sigma(A,B,A3,B3)) ) ) ) ) ).

% sum.Sigma
tff(fact_7055_prod_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),B3: fun(A,set(B)),G: fun(A,fun(B,C))] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B3,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_adr(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B3),G)),A3) = 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),product_case_prod(A,B,C,G)),product_Sigma(A,B,A3,B3)) ) ) ) ) ).

% prod.Sigma
tff(fact_7056_Func__empty,axiom,
    ! [B: $tType,A: $tType,B3: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B3) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert2(fun(A,B)),aTP_Lamp_ads(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_7057_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: fun(A,set(B))] : product_Sigma(A,B,A3,B3) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),image2(A,set(product_prod(A,B)),aTP_Lamp_adu(fun(A,set(B)),fun(A,set(product_prod(A,B))),B3)),A3)) ).

% Sigma_def
tff(fact_7058_product__fold,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),B3)
       => ( product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_adw(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B3),bot_bot(set(product_prod(A,B))),A3) ) ) ) ).

% product_fold
tff(fact_7059_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: fun(A,B),Xaa: list(A),Y: A] :
          ( ( arg_min_list(A,B,Xb,Xaa) = Y )
         => ( ! [X3: A] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
               => ( Y != X3 ) )
           => ( ! [X3: A,Y3: 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),Y3),Zs2)) )
                 => ( Y != $let(
                        m2: A,
                        m2:= arg_min_list(A,B,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),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) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_7060_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ( ~ aa(set(B),$o,finite_finite2(B),B3)
       => ~ aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3))) ) ) ).

% infinite_cartesian_product
tff(fact_7061_to__nat__on__def,axiom,
    ! [A: $tType,S2: set(A)] : countable_to_nat_on(A,S2) = fChoice(fun(A,nat),aTP_Lamp_adx(set(A),fun(fun(A,nat),$o),S2)) ).

% to_nat_on_def
tff(fact_7062_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_fj(nat,fun(nat,fun(nat,$o)),Mb))) = product_Sigma(nat,nat,aa(nat,set(nat),set_ord_atMost(nat),Mb),aTP_Lamp_ady(nat,fun(nat,set(nat)),Mb)) ).

% pairs_le_eq_Sigma
tff(fact_7063_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_7064_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R2: set(product_prod(A,A)),As3: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R2,As3)
        <=> ! [I4: A,J3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,I4),J3),R2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As3,I4)),aa(A,B,As3,J3)) ) ) ) ).

% relChain_def
tff(fact_7065_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,ord_less(nat))) ).

% natLess_def
tff(fact_7066_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A3,F2)),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)))
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A3)),B3) ) ).

% Gr_incl
tff(fact_7067_list__encode_Opelims,axiom,
    ! [Xb: list(nat),Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( accp(list(nat),nat_list_encode_rel,Xb)
       => ( ( ( Xb = nil(nat) )
           => ( ( Y = 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) )
               => ( ( Y = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),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_7068_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Y: A] :
          ( ( min_list(A,Xb) = Y )
         => ( 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) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_acy(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) )
                 => ( ( Y = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_7069_remdups__adj_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A)] :
      ( ( remdups_adj(A,Xb) = Y )
     => ( accp(list(A),remdups_adj_rel(A),Xb)
       => ( ( ( Xb = nil(A) )
           => ( ( Y = 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)) )
               => ( ( Y = 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,Y3: 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),Y3),Xs2)) )
                 => ( ( Y = $ite(X3 = Y3,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),Y3),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),Y3),Xs2))) ) ) ) ) ) ) ).

% remdups_adj.pelims
tff(fact_7070_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,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb)),aTP_Lamp_adz(nat,fun(nat,set(nat)),Nb))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Nb))) ).

% Restr_natLeq
tff(fact_7071_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_7072_remdups__upt,axiom,
    ! [Mb: nat,Nb: nat] : remdups(nat,upt(Mb,Nb)) = upt(Mb,Nb) ).

% remdups_upt
tff(fact_7073_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_7074_hd__upt,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( aa(list(nat),nat,hd(nat),upt(Ia,J)) = Ia ) ) ).

% hd_upt
tff(fact_7075_drop__upt,axiom,
    ! [Mb: nat,Ia: nat,J: nat] : drop(nat,Mb,upt(Ia,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),Mb),J) ).

% drop_upt
tff(fact_7076_length__upt,axiom,
    ! [Ia: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(Ia,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Ia) ).

% length_upt
tff(fact_7077_take__upt,axiom,
    ! [Ia: 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),Ia),Mb)),Nb)
     => ( take(nat,Mb,upt(Ia,Nb)) = upt(Ia,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),Mb)) ) ) ).

% take_upt
tff(fact_7078_upt__conv__Nil,axiom,
    ! [J: nat,Ia: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia)
     => ( upt(Ia,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_7079_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_7080_upt__eq__Nil__conv,axiom,
    ! [Ia: nat,J: nat] :
      ( ( upt(Ia,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia) ) ) ).

% upt_eq_Nil_conv
tff(fact_7081_nth__upt,axiom,
    ! [Ia: nat,K: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K)),J)
     => ( aa(nat,nat,nth(nat,upt(Ia,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),K) ) ) ).

% nth_upt
tff(fact_7082_distinct__upt,axiom,
    ! [Ia: nat,J: nat] : distinct(nat,upt(Ia,J)) ).

% distinct_upt
tff(fact_7083_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_7084_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_7085_atLeastLessThan__upt,axiom,
    ! [Ia: nat,J: nat] : set_or7035219750837199246ssThan(nat,Ia,J) = aa(list(nat),set(nat),set2(nat),upt(Ia,J)) ).

% atLeastLessThan_upt
tff(fact_7086_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_7087_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_7088_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_7089_upt__0,axiom,
    ! [Ia: nat] : upt(Ia,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_7090_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_7091_upt__conv__Cons,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( upt(Ia,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Ia),upt(aa(nat,nat,suc,Ia),J)) ) ) ).

% upt_conv_Cons
tff(fact_7092_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_7093_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,ord_less_eq(nat))) ).

% natLeq_def
tff(fact_7094_upt__add__eq__append,axiom,
    ! [Ia: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( upt(Ia,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(Ia,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_7095_upt__eq__Cons__conv,axiom,
    ! [Ia: nat,J: nat,Xb: nat,Xs: list(nat)] :
      ( ( upt(Ia,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),Ia),J)
        & ( Ia = Xb )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ia),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_7096_upt__rec,axiom,
    ! [Ia: nat,J: nat] :
      upt(Ia,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Ia),upt(aa(nat,nat,suc,Ia),J)),nil(nat)) ).

% upt_rec
tff(fact_7097_upt__Suc,axiom,
    ! [Ia: nat,J: nat] :
      upt(Ia,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J),aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(Ia,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_7098_upt__Suc__append,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
     => ( upt(Ia,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(Ia,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_7099_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_aeb(nat,fun(nat,set(nat)),Nb))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Nb))) ).

% Restr_natLeq2
tff(fact_7100_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_7101_map__ident,axiom,
    ! [A: $tType,X: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_lg(A,A)),X) = X ).

% map_ident
tff(fact_7102_list_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),A2) = nil(A) )
    <=> ( A2 = nil(B) ) ) ).

% list.map_disc_iff
tff(fact_7103_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),map(B,A,F2),Xs) )
    <=> ( Xs = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_7104_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = nil(A) )
    <=> ( Xs = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_7105_list_Omap__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,A),F2: fun(C,B),V2: list(C)] : aa(list(B),list(A),map(B,A,G),aa(list(C),list(B),map(C,B,F2),V2)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,G),F2)),V2) ).

% list.map_comp
tff(fact_7106_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),List: list(C)] : aa(list(B),list(A),map(B,A,F2),aa(list(C),list(B),map(C,B,G),List)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F2),G)),List) ).

% List.map.compositionality
tff(fact_7107_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),Xs: list(C)] : aa(list(B),list(A),map(B,A,F2),aa(list(C),list(B),map(C,B,G),Xs)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F2),G)),Xs) ).

% map_map
tff(fact_7108_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(B),list(A),map(B,A,G),Xs) )
    <=> ! [X2: B] :
          ( member(B,X2,aa(list(B),set(B),set2(B),Xs))
         => ( aa(B,A,F2,X2) = aa(B,A,G,X2) ) ) ) ).

% map_eq_conv
tff(fact_7109_length__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_7110_map__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F2),Xs)),aa(list(B),list(A),map(B,A,F2),Ys)) ).

% map_append
tff(fact_7111_map__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Nb: nat,Xb: B] : aa(list(B),list(A),map(B,A,F2),replicate(B,Nb,Xb)) = replicate(A,Nb,aa(B,A,F2,Xb)) ).

% map_replicate
tff(fact_7112_list_Oset__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),V2: list(B)] : aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),V2)) = aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),V2)) ).

% list.set_map
tff(fact_7113_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,F2),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_map_eq_map
tff(fact_7114_map__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: A,Xs: list(A),F2: fun(A,B),V2: B] :
      ( ~ member(A,Y,aa(list(A),set(A),set2(A),Xs))
     => ( aa(list(A),list(B),map(A,B,fun_upd(A,B,F2,Y,V2)),Xs) = aa(list(A),list(B),map(A,B,F2),Xs) ) ) ).

% map_fun_upd
tff(fact_7115_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),comp(list(C),list(B),list(A),map(C,B,F2)),map(A,C,G)) = map(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F2),G)) ).

% map_comp_map
tff(fact_7116_List_Omap_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),comp(list(C),list(B),list(A),map(C,B,F2)),map(A,C,G)) = map(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F2),G)) ).

% List.map.comp
tff(fact_7117_size__list__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F2),aa(list(B),list(A),map(B,A,G),Xs)) = aa(list(B),nat,size_list(B,aa(fun(B,A),fun(B,nat),comp(A,nat,B,F2),G)),Xs) ).

% size_list_map
tff(fact_7118_list_Omap__id0,axiom,
    ! [A: $tType] : map(A,A,id(A)) = id(list(A)) ).

% list.map_id0
tff(fact_7119_nth__map,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F2: fun(A,B)] :
      ( aa(nat,$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,F2),Xs)),Nb) = aa(A,B,F2,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% nth_map
tff(fact_7120_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_aec(fun(B,A),fun(B,list(A)),F2)),Xs)) = aa(list(B),list(A),map(B,A,F2),Xs) ).

% concat_map_singleton
tff(fact_7121_inj__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A))))
    <=> inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_map
tff(fact_7122_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_7123_enumerate__map__upt,axiom,
    ! [A: $tType,Nb: nat,F2: fun(nat,A),Mb: nat] : enumerate(A,Nb,aa(list(nat),list(A),map(nat,A,F2),upt(Nb,Mb))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_aed(fun(nat,A),fun(nat,product_prod(nat,A)),F2)),upt(Nb,Mb)) ).

% enumerate_map_upt
tff(fact_7124_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F2: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F2),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G),Ys))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,product_case_prod(B,D,A,aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_aee(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F2),G))),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_7125_zip__map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,F2: fun(C,A),Xs: list(C),G: fun(D,B),Ys: list(D)] : zip(A,B,aa(list(C),list(A),map(C,A,F2),Xs),aa(list(D),list(B),map(D,B,G),Ys)) = aa(list(product_prod(C,D)),list(product_prod(A,B)),map(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_aef(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),G))),zip(C,D,Xs,Ys)) ).

% zip_map_map
tff(fact_7126_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F2: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F2),zip(B,C,aa(list(D),list(B),map(D,B,G),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,product_case_prod(D,C,A,aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_aeg(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F2),G))),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_7127_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F2: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F2),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),product_case_prod(A,C,product_prod(A,B),aTP_Lamp_aeh(fun(C,B),fun(A,fun(C,product_prod(A,B))),F2))),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_7128_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F2),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),product_case_prod(C,B,product_prod(A,B),aTP_Lamp_aei(fun(C,A),fun(C,fun(B,product_prod(A,B))),F2))),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_7129_list_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(list(B),list(A),map(B,A,F2),nil(B)) = nil(A) ).

% list.simps(8)
tff(fact_7130_list_Omap__sel_I2_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F2: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),list(B),tl(B),aa(list(A),list(B),map(A,B,F2),A2)) = aa(list(A),list(B),map(A,B,F2),aa(list(A),list(A),tl(A),A2)) ) ) ).

% list.map_sel(2)
tff(fact_7131_list_Omap__sel_I1_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F2: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F2),A2)) = aa(A,B,F2,aa(list(A),A,hd(A),A2)) ) ) ).

% list.map_sel(1)
tff(fact_7132_hd__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B)] :
      ( ( Xs != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F2),Xs)) = aa(A,B,F2,aa(list(A),A,hd(A),Xs)) ) ) ).

% hd_map
tff(fact_7133_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F2: fun(D,B),Xs: list(D),G: fun(D,C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,product_case_prod(B,C,A,H)),zip(B,C,aa(list(D),list(B),map(D,B,F2),Xs),aa(list(D),list(C),map(D,C,G),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_aej(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F2),G)),Xs) ).

% map2_map_map
tff(fact_7134_drop__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F2: fun(B,A),Xs: list(B)] : drop(A,Nb,aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),drop(B,Nb,Xs)) ).

% drop_map
tff(fact_7135_map__injective,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(B),list(A),map(B,A,F2),Ys) )
     => ( inj_on(B,A,F2,top_top(set(B)))
       => ( Xs = Ys ) ) ) ).

% map_injective
tff(fact_7136_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_7137_List_Omap_Oidentity,axiom,
    ! [A: $tType] : map(A,A,aTP_Lamp_lg(A,A)) = id(list(A)) ).

% List.map.identity
tff(fact_7138_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),nat),comp(list(B),nat,list(A),size_list(B,F2)),map(A,B,G)) = size_list(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F2),G)) ).

% list.size_gen_o_map
tff(fact_7139_list_Omap__ident,axiom,
    ! [A: $tType,Ta: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_lg(A,A)),Ta) = Ta ).

% list.map_ident
tff(fact_7140_map__concat,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(list(B))] : aa(list(B),list(A),map(B,A,F2),concat(B,Xs)) = concat(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),Xs)) ).

% map_concat
tff(fact_7141_rotate1__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate1(A),aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),rotate1(B),Xs)) ).

% rotate1_map
tff(fact_7142_remdups__map__remdups,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : remdups(A,aa(list(B),list(A),map(B,A,F2),remdups(B,Xs))) = remdups(A,aa(list(B),list(A),map(B,A,F2),Xs)) ).

% remdups_map_remdups
tff(fact_7143_nths__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),I5: set(nat)] : nths(A,aa(list(B),list(A),map(B,A,F2),Xs),I5) = aa(list(B),list(A),map(B,A,F2),nths(B,Xs,I5)) ).

% nths_map
tff(fact_7144_take__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F2: fun(B,A),Xs: list(B)] : take(A,Nb,aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),take(B,Nb,Xs)) ).

% take_map
tff(fact_7145_map__update,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),K: nat,Y: B] : aa(list(B),list(A),map(B,A,F2),list_update(B,Xs,K,Y)) = list_update(A,aa(list(B),list(A),map(B,A,F2),Xs),K,aa(B,A,F2,Y)) ).

% map_update
tff(fact_7146_map__tl,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),tl(B),Xs)) = aa(list(A),list(A),tl(A),aa(list(B),list(A),map(B,A,F2),Xs)) ).

% map_tl
tff(fact_7147_rotate__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate(A,Nb),aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),rotate(B,Nb),Xs)) ).

% rotate_map
tff(fact_7148_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ? [Z6: B,Zs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z6),Zs3) )
          & ( aa(B,A,F2,Z6) = Y )
          & ( aa(list(B),list(A),map(B,A,F2),Zs3) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_7149_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),F2: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = aa(list(B),list(A),map(B,A,F2),Ys) )
    <=> ? [Z6: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z6),Zs3) )
          & ( Xb = aa(B,A,F2,Z6) )
          & ( Xs = aa(list(B),list(A),map(B,A,F2),Zs3) ) ) ) ).

% Cons_eq_map_conv
tff(fact_7150_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),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,F2,Z2) = Y )
          & ( aa(list(B),list(A),map(B,A,F2),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_7151_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),F2: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = aa(list(B),list(A),map(B,A,F2),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,F2,Z2) )
          & ( Xs = aa(list(B),list(A),map(B,A,F2),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_7152_list_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X21: B,X222: list(B)] : aa(list(B),list(A),map(B,A,F2),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,F2,X21)),aa(list(B),list(A),map(B,A,F2),X222)) ).

% list.simps(9)
tff(fact_7153_append__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Ys: list(A),Zs: list(A),F2: fun(B,A),Xs: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(B),list(A),map(B,A,F2),Xs) )
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( Ys = aa(list(B),list(A),map(B,A,F2),Us2) )
          & ( Zs = aa(list(B),list(A),map(B,A,F2),Vs3) ) ) ) ).

% append_eq_map_conv
tff(fact_7154_map__eq__append__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( Ys = aa(list(B),list(A),map(B,A,F2),Us2) )
          & ( Zs = aa(list(B),list(A),map(B,A,F2),Vs3) ) ) ) ).

% map_eq_append_conv
tff(fact_7155_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aTP_Lamp_lu(A,fun(B,A),K)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_7156_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys: list(B),F2: fun(A,B)] :
      ( ? [Xs3: list(A)] : Ys = aa(list(A),list(B),map(A,B,F2),Xs3)
    <=> ! [X2: B] :
          ( member(B,X2,aa(list(B),set(B),set2(B),Ys))
         => ? [Xa3: A] : X2 = aa(A,B,F2,Xa3) ) ) ).

% ex_map_conv
tff(fact_7157_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) )
       => ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,G),Ys) ) ) ) ).

% map_cong
tff(fact_7158_map__idI,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,A,F2,X3) = X3 ) )
     => ( aa(list(A),list(A),map(A,A,F2),Xs) = Xs ) ) ).

% map_idI
tff(fact_7159_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) )
     => ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,G),Xs) ) ) ).

% map_ext
tff(fact_7160_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Xaa: list(A),F2: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z2: A,Za: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
         => ( member(A,Za,aa(list(A),set(A),set2(A),Xaa))
           => ( ( aa(A,B,F2,Z2) = aa(A,B,Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F2),Xb) = aa(list(A),list(B),map(A,B,Fa),Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_7161_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z2: A] :
          ( member(A,Z2,aa(list(A),set(A),set2(A),Xb))
         => ( aa(A,B,F2,Z2) = aa(A,B,G,Z2) ) )
     => ( aa(list(A),list(B),map(A,B,F2),Xb) = aa(list(A),list(B),map(A,B,G),Xb) ) ) ).

% list.map_cong0
tff(fact_7162_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xb = Ya )
     => ( ! [Z2: A] :
            ( member(A,Z2,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,B,F2,Z2) = aa(A,B,G,Z2) ) )
       => ( aa(list(A),list(B),map(A,B,F2),Xb) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong
tff(fact_7163_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(C),list(A),map(C,A,G),Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_7164_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(B),list(A),map(B,A,F2),Ys) )
     => ( inj_on(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
       => ( Xs = Ys ) ) ) ).

% map_inj_on
tff(fact_7165_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
     => ( ( aa(list(A),list(B),map(A,B,F2),Xs) = aa(list(A),list(B),map(A,B,F2),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_on_map_eq_map
tff(fact_7166_image__set,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),Xs)) ).

% image_set
tff(fact_7167_distinct__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xs))
    <=> ( distinct(B,Xs)
        & inj_on(B,A,F2,aa(list(B),set(B),set2(B),Xs)) ) ) ).

% distinct_map
tff(fact_7168_natLeq__underS__less,axiom,
    ! [Nb: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,Nb) = collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb)) ).

% natLeq_underS_less
tff(fact_7169_remdups__adj__map__injective,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( remdups_adj(B,aa(list(A),list(B),map(A,B,F2),Xs)) = aa(list(A),list(B),map(A,B,F2),remdups_adj(A,Xs)) ) ) ).

% remdups_adj_map_injective
tff(fact_7170_map__removeAll__inj,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xb: A,Xs: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(list(A),list(B),map(A,B,F2),aa(list(A),list(A),removeAll(A,Xb),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F2,Xb)),aa(list(A),list(B),map(A,B,F2),Xs)) ) ) ).

% map_removeAll_inj
tff(fact_7171_map__replicate__trivial,axiom,
    ! [A: $tType,Xb: A,Ia: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_aek(A,fun(nat,A),Xb)),upt(zero_zero(nat),Ia)) = replicate(A,Ia,Xb) ).

% map_replicate_trivial
tff(fact_7172_distinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xb: B,Xs: list(B)] :
          ( distinct(A,aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xb),Xs)))
        <=> ( ~ member(A,aa(B,A,F2,Xb),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs)))
            & distinct(A,aa(list(B),list(A),map(B,A,F2),Xs)) ) ) ) ).

% distinct_insort_key
tff(fact_7173_map__removeAll__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xb: A,Xs: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(B),map(A,B,F2),aa(list(A),list(A),removeAll(A,Xb),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F2,Xb)),aa(list(A),list(B),map(A,B,F2),Xs)) ) ) ).

% map_removeAll_inj_on
tff(fact_7174_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),Nb: nat] : aa(list(nat),list(A),map(nat,A,F2),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,F2,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_ael(fun(nat,A),fun(nat,A),F2)),upt(zero_zero(nat),Nb))) ).

% map_upt_Suc
tff(fact_7175_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_7176_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A))))
     => inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_mapD
tff(fact_7177_nth__map__upt,axiom,
    ! [A: $tType,Ia: nat,Nb: nat,Mb: nat,F2: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),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,F2),upt(Mb,Nb))),Ia) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Ia)) ) ) ).

% nth_map_upt
tff(fact_7178_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F2: 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,F2),Xs))),X) = $ite(member(A,X,aa(list(A),set(A),set2(A),Xs)),aa(B,option(B),some(B),aa(A,B,F2,X)),none(B)) ).

% map_of_zip_map
tff(fact_7179_inj__on__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(list(A))] :
      ( inj_on(A,B,F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),A3)))
     => inj_on(list(A),list(B),map(A,B,F2),A3) ) ).

% inj_on_mapI
tff(fact_7180_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Mb: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I2)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F2),upt(Mb,Nb)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_7181_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( Nb = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = Nb ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_aen(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),Nb)) ) ) ) ).

% transpose_rectangle
tff(fact_7182_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_7183_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_7184_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_7185_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_7186_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_7187_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_7188_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_7189_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_7190_List_Obind__def,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F2: fun(B,list(A))] : bind(B,A,Xs,F2) = concat(A,aa(list(B),list(list(A)),map(B,list(A),F2),Xs)) ).

% List.bind_def
tff(fact_7191_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),Xs)) = aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),transpose(B,Xs)) ).

% transpose_map_map
tff(fact_7192_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_aeo(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_7193_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_aep(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,Nb,Xs))) ).

% n_lists.simps(2)
tff(fact_7194_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_aeq(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_7195_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_aer(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_7196_transpose_Oelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xb) = Y )
     => ( ( ( Xb = nil(list(A)) )
         => ( Y != 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) )
             => ( Y != 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) )
               => ( Y != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aes(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_aet(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_7197_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_aes(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_aet(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_7198_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_7199_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_7200_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).

% transpose.simps(1)
tff(fact_7201_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_7202_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_7203_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_7204_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_aeu(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_7205_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)),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C)),aTP_Lamp_aew(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_7206_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)),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C)),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_aex(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_7207_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),product_case_prod(B,A,product_prod(A,B),aTP_Lamp_aey(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_7208_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),product_Pair(A,B,Xb)),take(B,Nb,Ys)) ).

% zip_replicate1
tff(fact_7209_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_7210_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_7211_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_xj(nat,nat)),upt(aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_7212_transpose__empty,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( transpose(A,Xs) = nil(list(A)) )
    <=> ! [X2: list(A)] :
          ( member(list(A),X2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( X2 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_7213_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Nb: nat,Y: B] : zip(A,B,Xs,replicate(B,Nb,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_aey(B,fun(A,product_prod(A,B))),Y)),take(A,Nb,Xs)) ).

% zip_replicate2
tff(fact_7214_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_7215_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_aer(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_7216_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),product_Pair(A,B,Xb)),Ys)),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_7217_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_7218_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_aez(fun(A,B),fun(A,product_prod(A,B)),F2)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F2),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_7219_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_afa(fun(A,option(B)),fun(A,product_prod(A,B)),Mb)),Xs)) = Mb ) ) ).

% map_of_map_keys
tff(fact_7220_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_afb(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_7221_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_7222_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_aes(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_aet(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_7223_transpose_Opelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xb) = Y )
     => ( accp(list(list(A)),transpose_rel(A),Xb)
       => ( ( ( Xb = nil(list(A)) )
           => ( ( Y = 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) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) ) )
           => ~ ! [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) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aes(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_aet(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_7224_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_7225_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_7226_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),$o)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => aa(list(list(A)),$o,P,nil(list(A))) )
       => ( ! [Xss2: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))
             => ( aa(list(list(A)),$o,P,Xss2)
               => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) ) )
         => ( ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))
               => ( aa(list(list(A)),$o,P,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_aet(A,fun(list(A),list(list(A)))))),Xss2))))
                 => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2)) ) )
           => aa(list(list(A)),$o,P,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_7227_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_aeq(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_7228_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_7229_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aet(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afc(list(A),$o)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_7230_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_afd(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F2)),X5) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_7231_filter__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),filter2(A,Q),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afe(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),Xs) ).

% filter_filter
tff(fact_7232_filter__True,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) )
     => ( aa(list(A),list(A),filter2(A,P),Xs) = Xs ) ) ).

% filter_True
tff(fact_7233_filter__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,P),Xs)),aa(list(A),list(A),filter2(A,P),Ys)) ).

% filter_append
tff(fact_7234_remove1__filter__not,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      ( ~ aa(A,$o,P,Xb)
     => ( remove1(A,Xb,aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% remove1_filter_not
tff(fact_7235_removeAll__filter__not,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      ( ~ aa(A,$o,P,Xb)
     => ( aa(list(A),list(A),removeAll(A,Xb),aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% removeAll_filter_not
tff(fact_7236_set__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs)) = collect(A,aa(list(A),fun(A,$o),aTP_Lamp_aff(fun(A,$o),fun(list(A),fun(A,$o)),P),Xs)) ).

% set_filter
tff(fact_7237_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A,nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_7238_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( groups8242544230860333062m_list(A,Ns) = zero_zero(A) )
        <=> ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Ns))
             => ( X2 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_7239_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_7240_filter__False,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X3) )
     => ( aa(list(A),list(A),filter2(A,P),Xs) = nil(A) ) ) ).

% filter_False
tff(fact_7241_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_7242_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_afg(B,A)),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_7243_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F2),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,P),F2)),Xs)) ).

% length_filter_map
tff(fact_7244_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_bk(nat,nat)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ) ).

% sum_list_upt
tff(fact_7245_filter_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(list(A),list(A),filter2(A,P),nil(A)) = nil(A) ).

% filter.simps(1)
tff(fact_7246_filter__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Nb: nat,Xb: A] :
      aa(list(A),list(A),filter2(A,P),replicate(A,Nb,Xb)) = $ite(aa(A,$o,P,Xb),replicate(A,Nb,Xb),nil(A)) ).

% filter_replicate
tff(fact_7247_filter__insort__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),Xb: A,F2: fun(A,B),Xs: list(A)] :
          ( ~ aa(A,$o,P,Xb)
         => ( aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ) ).

% filter_insort_triv
tff(fact_7248_removeAll__filter__not__eq,axiom,
    ! [A: $tType,Xb: A] : removeAll(A,Xb) = filter2(A,aTP_Lamp_afh(A,fun(A,$o),Xb)) ).

% removeAll_filter_not_eq
tff(fact_7249_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_7250_partition__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] : member(list(A),Xs,shuffles(A,aa(list(A),list(A),filter2(A,P),Xs),aa(list(A),list(A),filter2(A,aTP_Lamp_be(fun(A,$o),fun(A,$o),P)),Xs))) ).

% partition_in_shuffles
tff(fact_7251_filter_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P,Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),filter2(A,P),Xs)),aa(list(A),list(A),filter2(A,P),Xs)) ).

% filter.simps(2)
tff(fact_7252_filter__shuffles,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] : aa(set(list(A)),set(list(A)),image2(list(A),list(A),filter2(A,P)),shuffles(A,Xs,Ys)) = shuffles(A,aa(list(A),list(A),filter2(A,P),Xs),aa(list(A),list(A),filter2(A,P),Ys)) ).

% filter_shuffles
tff(fact_7253_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_7254_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_7255_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_7256_empty__filter__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),filter2(A,P),Xs) )
    <=> ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X2) ) ) ).

% empty_filter_conv
tff(fact_7257_filter__empty__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xs) = nil(A) )
    <=> ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X2) ) ) ).

% filter_empty_conv
tff(fact_7258_length__filter__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_7259_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,$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,P),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aTP_Lamp_be(fun(A,$o),fun(A,$o),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_7260_filter__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : filter3(A,P,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs)) ).

% filter_set
tff(fact_7261_filter__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,P,X3)
            <=> aa(A,$o,Q,X3) ) )
       => ( aa(list(A),list(A),filter2(A,P),Xs) = aa(list(A),list(A),filter2(A,Q),Ys) ) ) ) ).

% filter_cong
tff(fact_7262_filter__id__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xs) = Xs )
    <=> ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) ) ) ).

% filter_id_conv
tff(fact_7263_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_7264_Cons__eq__filterD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$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,P),Ys) )
     => ? [Us3: list(A),Vs2: 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),Vs2)) )
          & ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P,X) )
          & aa(A,$o,P,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs2) ) ) ) ).

% Cons_eq_filterD
tff(fact_7265_filter__eq__ConsD,axiom,
    ! [A: $tType,P: fun(A,$o),Ys: list(A),Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) )
     => ? [Us3: list(A),Vs2: 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),Vs2)) )
          & ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P,X) )
          & aa(A,$o,P,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs2) ) ) ) ).

% filter_eq_ConsD
tff(fact_7266_Cons__eq__filter__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$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,P),Ys) )
    <=> ? [Us2: list(A),Vs3: 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),Vs3)) )
          & ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X2) )
          & aa(A,$o,P,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs3) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_7267_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Ys: list(A),Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) )
    <=> ? [Us2: list(A),Vs3: 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),Vs3)) )
          & ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X2) )
          & aa(A,$o,P,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs3) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_7268_remdups__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : remdups(A,aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),remdups(A,Xs)) ).

% remdups_filter
tff(fact_7269_inter__set__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_a(set(A),fun(A,$o),A3)),Xs)) ).

% inter_set_filter
tff(fact_7270_distinct__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% distinct_filter
tff(fact_7271_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),aa(list(A),list(A),filter2(A,P),Xs)))),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs))) ).

% sum_list_filter_le_nat
tff(fact_7272_distinct__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),P: fun(B,$o)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xs))
     => distinct(A,aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xs))) ) ).

% distinct_map_filter
tff(fact_7273_length__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,Xss)) = groups8242544230860333062m_list(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss)) ).

% length_concat
tff(fact_7274_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(A),P: fun(A,$o),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => ( ~ aa(A,$o,P,X3)
               => ( aa(A,B,F2,X3) = zero_zero(B) ) ) )
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),aa(list(A),list(A),filter2(A,P),Xs))) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_7275_filter__concat,axiom,
    ! [A: $tType,P3: fun(A,$o),Xs: list(list(A))] : aa(list(A),list(A),filter2(A,P3),concat(A,Xs)) = concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),filter2(A,P3)),Xs)) ).

% filter_concat
tff(fact_7276_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(B,A),G: 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_bv(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xs))) ) ).

% sum_list_addf
tff(fact_7277_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F2: fun(B,A),P: fun(B,$o),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_afi(fun(B,A),fun(fun(B,$o),fun(B,A)),F2),P)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_7278_filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P),F2)),Xs)) ).

% filter_map
tff(fact_7279_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_afj(fun(A,B),fun(A,fun(A,$o)),F2),Y)),Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Y)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_7280_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_7281_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) )
          <=> ! [X2: A] :
                ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
               => ( X2 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_7282_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_7283_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_7284_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_afk(list(A),fun(set(nat),fun(A,$o)),Xs),S)),Xs) = nths(A,Xs,S) ) ) ).

% filter_in_nths
tff(fact_7285_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xs))) ) ) ).

% sum_list_mono
tff(fact_7286_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_afl(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_7287_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_afm(A,fun(A,$o)),Y)),Xs)) ).

% set_minus_filter_out
tff(fact_7288_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_afn(list(A),fun(A,$o),Ys)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_7289_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_afo(list(A),fun(A,$o),Ys)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_7290_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_afn(list(A),fun(A,$o),Xs)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_7291_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_afo(list(A),fun(A,$o),Xs)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_7292_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),Xs) = nths(A,Xs,collect(nat,aa(list(A),fun(nat,$o),aTP_Lamp_afp(fun(A,$o),fun(list(A),fun(nat,$o)),P),Xs))) ).

% filter_eq_nths
tff(fact_7293_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,$o),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P3),Xs)) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(list(A),fun(nat,$o),aTP_Lamp_afp(fun(A,$o),fun(list(A),fun(nat,$o)),P3),Xs))) ).

% length_filter_conv_card
tff(fact_7294_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xs))) ) ) ) ).

% sum_list_strict_mono
tff(fact_7295_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),groups8242544230860333062m_list(A,Ns)) ) ) ).

% elem_le_sum_list
tff(fact_7296_sum_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(A),G: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(list(A),set(A),set2(A),Xs)) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xs)) ) ) ) ).

% sum.distinct_set_conv_list
tff(fact_7297_sum__list__distinct__conv__sum__set,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(A),F2: fun(A,B)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xs)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% sum_list_distinct_conv_sum_set
tff(fact_7298_sum__list__map__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xb: A,Xs: list(A),F2: 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,F2),Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xb)),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),remove1(A,Xb,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_7299_sum__code,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(list(B),set(B),set2(B),Xs)) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),remdups(B,Xs))) ) ).

% sum_code
tff(fact_7300_size__list__conv__sum__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : aa(list(A),nat,size_list(A,F2),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% size_list_conv_sum_list
tff(fact_7301_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R2: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_afq(A,fun(B,A),R2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xs))),R2) ) ).

% sum_list_triv
tff(fact_7302_sum__list__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_jo(fun(A,nat),fun(A,nat),F2)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_7303_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs)) = aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),collect(A,P)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_7304_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_7305_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)),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)),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)),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_7306_card__length__sum__list,axiom,
    ! [Mb: nat,N3: nat] : aa(set(list(nat)),nat,finite_card(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_7307_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_afu(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_7308_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
         => ( groups8242544230860333062m_list(A,list_update(A,Xs,K,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),K)) ) ) ) ).

% sum_list_update
tff(fact_7309_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aes(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afc(list(A),$o)),Xss)) ).

% transpose_aux_filter_head
tff(fact_7310_nth__transpose,axiom,
    ! [A: $tType,Ia: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),Ia) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_afv(nat,fun(list(A),A),Ia)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afw(nat,fun(list(A),$o),Ia)),Xs)) ) ) ).

% nth_transpose
tff(fact_7311_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_afx(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afc(list(A),$o)),Xs)) ).

% transpose_max_length
tff(fact_7312_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,F2,Xs) = aa(list(B),list(A),map(B,A,aa(fun(B,option(A)),fun(B,A),comp(option(A),A,B,the2(A)),F2)),aa(list(B),list(B),filter2(B,aTP_Lamp_afy(fun(B,option(A)),fun(B,$o),F2)),Xs)) ).

% map_filter_def
tff(fact_7313_foldr__append,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Xs: list(B),Ys: list(B),A2: A] : aa(A,A,foldr(B,A,F2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),A2) = aa(A,A,foldr(B,A,F2,Xs),aa(A,A,foldr(B,A,F2,Ys),A2)) ).

% foldr_append
tff(fact_7314_foldr__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Nb: nat,Xb: B] : foldr(B,A,F2,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),F2,Xb)) ).

% foldr_replicate
tff(fact_7315_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F2: fun(C,B),Xs: list(C),A2: A] : aa(A,A,foldr(B,A,G,aa(list(C),list(B),map(C,B,F2),Xs)),A2) = aa(A,A,foldr(C,A,aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G),F2),Xs),A2) ).

% foldr_map
tff(fact_7316_foldr__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xb: B,Xs: list(B)] : foldr(B,A,F2,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),F2,Xb)),foldr(B,A,F2,Xs)) ).

% foldr_Cons
tff(fact_7317_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A4: A,X3: B] :
              ( member(B,X3,aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F2,X3),A4) = aa(A,A,aa(B,fun(A,A),G,X3),A4) ) )
         => ( aa(A,A,foldr(B,A,F2,L),A2) = aa(A,A,foldr(B,A,G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_7318_foldr__Nil,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A))] : foldr(B,A,F2,nil(B)) = id(A) ).

% foldr_Nil
tff(fact_7319_map__filter__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,option(A))] : map_filter(B,A,F2,nil(B)) = nil(A) ).

% map_filter_simps(2)
tff(fact_7320_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),P: fun(B,$o),Xs: list(B)] : foldr(B,A,F2,aa(list(B),list(B),filter2(B,P),Xs)) = foldr(B,A,aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_afz(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F2),P),Xs) ).

% foldr_filter
tff(fact_7321_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xb: B,Xs: list(B)] : map_filter(B,A,F2,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,F2,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_aga(fun(B,option(A)),fun(list(B),fun(A,list(A))),F2),Xs),aa(B,option(A),F2,Xb)) ).

% map_filter_simps(1)
tff(fact_7322_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_7323_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_agb(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A2),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_7324_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_afx(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_7325_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_agc(list(B),fun(nat,nat)),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_agd(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_age(list(B),$o)),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_7326_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,$o),Xs: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_agf(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F2),P),Xs) ).

% map_filter_map_filter
tff(fact_7327_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F2: fun(nat,A),Ns: list(nat)] :
          ( ! [X3: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Y3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,X3)),aa(nat,A,F2,Y3)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F2),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_7328_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss)),one_one(nat)) ).

% length_product_lists
tff(fact_7329_sorted__wrt__filter,axiom,
    ! [A: $tType,F2: fun(A,fun(A,$o)),Xs: list(A),P: fun(A,$o)] :
      ( sorted_wrt(A,F2,Xs)
     => sorted_wrt(A,F2,aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% sorted_wrt_filter
tff(fact_7330_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_agg(fun(list(A),A),fun(list(A),fun(A,$o)),G),Xs)),Xs)) ) ).

% sorted_same
tff(fact_7331_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),remove1(B,Xb,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_7332_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,P),Xs))) ) ) ).

% sorted_filter
tff(fact_7333_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xs))
        <=> sorted_wrt(B,aTP_Lamp_agh(fun(B,A),fun(B,fun(B,$o)),F2),Xs) ) ) ).

% sorted_map
tff(fact_7334_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xb: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xb),Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xs)) ) ) ).

% sorted_insort_key
tff(fact_7335_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,$o)),F2: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R,aa(list(B),list(A),map(B,A,F2),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_agi(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R),F2),Xs) ) ).

% sorted_wrt_map
tff(fact_7336_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A),Ia: nat,J: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),Ia)),aa(nat,A,nth(A,Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_7337_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_7338_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_7339_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : sorted_wrt(nat,ord_less(nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_7340_sorted__wrt1,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A] : sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) ).

% sorted_wrt1
tff(fact_7341_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_7342_sorted__wrt_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : sorted_wrt(A,P,nil(A)) ).

% sorted_wrt.simps(1)
tff(fact_7343_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_7344_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_agj(A,fun(A,$o)),Xs) ).

% sorted_wrt_true
tff(fact_7345_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_7346_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I5: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_7347_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_7348_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_7349_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_7350_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_7351_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_xr(A,A)),Xb),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_7352_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_7353_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_7354_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_7355_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_7356_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_7357_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_7358_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_7359_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))
        <=> ( ! [X2: A] :
                ( member(A,X2,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X2) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_7360_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X2: A] :
                ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
               => ! [Xa3: A] :
                    ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa3) ) ) ) ) ) ).

% sorted_append
tff(fact_7361_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => sorted_wrt(A,P,Xs) ) ).

% sorted_wrt01
tff(fact_7362_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_7363_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o)),Q: fun(A,fun(A,$o))] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,$o,aa(A,fun(A,$o),P,X3),Y3)
             => aa(A,$o,aa(A,fun(A,$o),Q,X3),Y3) ) ) )
     => ( sorted_wrt(A,P,Xs)
       => sorted_wrt(A,Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_7364_sorted__wrt__append,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( sorted_wrt(A,P,Xs)
        & sorted_wrt(A,P,Ys)
        & ! [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
           => ! [Xa3: A] :
                ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),P,X2),Xa3) ) ) ) ) ).

% sorted_wrt_append
tff(fact_7365_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))
        <=> ( ! [X2: A] :
                ( member(A,X2,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X2) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_7366_sorted__wrt_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A,Ys: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))
    <=> ( ! [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Ys))
           => aa(A,$o,aa(A,fun(A,$o),P,Xb),X2) )
        & sorted_wrt(A,P,Ys) ) ) ).

% sorted_wrt.simps(2)
tff(fact_7367_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) )
           => ( ! [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(3)
tff(fact_7368_sorted__wrt__take,axiom,
    ! [A: $tType,F2: fun(A,fun(A,$o)),Xs: list(A),Nb: nat] :
      ( sorted_wrt(A,F2,Xs)
     => sorted_wrt(A,F2,take(A,Nb,Xs)) ) ).

% sorted_wrt_take
tff(fact_7369_sorted__wrt__drop,axiom,
    ! [A: $tType,F2: fun(A,fun(A,$o)),Xs: list(A),Nb: nat] :
      ( sorted_wrt(A,F2,Xs)
     => sorted_wrt(A,F2,drop(A,Nb,Xs)) ) ).

% sorted_wrt_drop
tff(fact_7370_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_7371_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_7372_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_agk(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F2),G),Xs)),Xs))) ) ).

% sorted_map_same
tff(fact_7373_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_7374_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_7375_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_7376_sorted__wrt_Oelims_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A),Y: $o] :
      ( ( sorted_wrt(A,Xb,Xaa)
      <=> (Y) )
     => ( ( ( Xaa = nil(A) )
         => ~ (Y) )
       => ~ ! [X3: A,Ys3: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
             => ( (Y)
              <=> ~ ( ! [Xa3: A] :
                        ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys3))
                       => aa(A,$o,aa(A,fun(A,$o),Xb,X3),Xa3) )
                    & sorted_wrt(A,Xb,Ys3) ) ) ) ) ) ).

% sorted_wrt.elims(1)
tff(fact_7377_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ? [X3: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X3) = A3 )
              & sorted_wrt(A,ord_less_eq(A),X3)
              & distinct(A,X3)
              & ! [Y2: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y2) = A3 )
                    & sorted_wrt(A,ord_less_eq(A),Y2)
                    & distinct(A,Y2) )
                 => ( Y2 = X3 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_7378_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_7379_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,$o),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xs))
         => ( aa(B,$o,P,Xb)
           => ( aa(list(B),list(B),filter2(B,P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xb),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F2),Xb),aa(list(B),list(B),filter2(B,P),Xs)) ) ) ) ) ).

% filter_insort
tff(fact_7380_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_7381_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_xr(A,A)),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_7382_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_7383_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ~ ! [L3: list(A)] :
                ( sorted_wrt(A,ord_less(A),L3)
               => ( ( aa(list(A),set(A),set2(A),L3) = A3 )
                 => ( aa(list(A),nat,size_size(list(A)),L3) != aa(set(A),nat,finite_card(A),A3) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_7384_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ia: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
           => ( aa(nat,$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),Ia)),aa(nat,A,nth(A,Xs),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_7385_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_7386_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),Ia: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),aa(nat,nat,nth(nat,Ns),Ia)) ) ) ).

% sorted_wrt_less_idx
tff(fact_7387_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
                & aa(A,$o,P,X) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_agl(list(A),fun(fun(A,$o),fun(A,$o)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_7388_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A),Ys: list(A)] :
          ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F2),Xs))
           => ( distinct(B,aa(list(A),list(B),map(A,B,F2),Xs))
             => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F2),Ys))
               => ( distinct(B,aa(list(A),list(B),map(A,B,F2),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_7389_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A3 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A3) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_7390_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_xr(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_7391_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A2: A,Xs: list(A),F2: fun(A,B)] :
          ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F2),Xs))
           => ( ( aa(list(A),A,hd(A),aa(list(A),list(A),filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_agm(A,fun(fun(A,B),fun(A,$o)),A2),F2)),Xs)) = A2 )
             => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_7392_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),Ia: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
       => ( aa(nat,$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),Ia)),Xs)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),Ia)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),Ia) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_7393_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),Ia: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_afv(nat,fun(list(A),A),Ia)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afw(nat,fun(list(A),$o),Ia)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),Ia) ) ) ) ).

% transpose_column
tff(fact_7394_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_7395_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_7396_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_7397_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_7398_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_7399_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_7400_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_7401_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_7402_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_7403_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_7404_inj__on__rev,axiom,
    ! [A: $tType,A3: set(list(A))] : inj_on(list(A),list(A),rev(A),A3) ).

% inj_on_rev
tff(fact_7405_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_7406_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_7407_rev__eq__Cons__iff,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))) ) ) ).

% rev_eq_Cons_iff
tff(fact_7408_sorted__wrt__upto,axiom,
    ! [Ia: int,J: int] : sorted_wrt(int,ord_less(int),upto(Ia,J)) ).

% sorted_wrt_upto
tff(fact_7409_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),rev(A),Xs))
    <=> sorted_wrt(A,aTP_Lamp_agn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P),Xs) ) ).

% sorted_wrt_rev
tff(fact_7410_sorted__upto,axiom,
    ! [Mb: int,Nb: int] : sorted_wrt(int,ord_less_eq(int),upto(Mb,Nb)) ).

% sorted_upto
tff(fact_7411_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_7412_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_7413_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_7414_rev_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rev(A),nil(A)) = nil(A) ).

% rev.simps(1)
tff(fact_7415_rev__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),rev(B),Xs)) ).

% rev_map
tff(fact_7416_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_7417_rev__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),rev(A),Xs)) ).

% rev_filter
tff(fact_7418_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_7419_rev__drop,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),drop(A,Ia,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Ia),aa(list(A),list(A),rev(A),Xs)) ).

% rev_drop
tff(fact_7420_rev__take,axiom,
    ! [A: $tType,Ia: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),take(A,Ia,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Ia),aa(list(A),list(A),rev(A),Xs)) ).

% rev_take
tff(fact_7421_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_7422_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_7423_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_7424_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),rev(A),list_update(A,Xs,K,Y)) = list_update(A,aa(list(A),list(A),rev(A),Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_7425_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_7426_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_7427_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_7428_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ia: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ia),J)
           => ( aa(nat,$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),Ia)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_7429_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y) = $ite(Xs = nil(A),Y,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_7430_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_7431_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),Ia: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afw(nat,fun(list(A),$o),Ia)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),Ia)) ) ) ) ).

% transpose_column_length
tff(fact_7432_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => ( aa(set(B),$o,finite_finite2(B),A3)
         => ~ ! [L3: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),L3))
               => ( ( aa(list(B),set(B),set2(B),L3) = A3 )
                 => ( aa(list(B),nat,size_size(list(B)),L3) != aa(set(B),nat,finite_card(B),A3) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_7433_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_afc(list(A),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_7434_takeWhile__idem,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : takeWhile(A,P,takeWhile(A,P,Xs)) = takeWhile(A,P,Xs) ).

% takeWhile_idem
tff(fact_7435_takeWhile__eq__all__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( takeWhile(A,P,Xs) = Xs )
    <=> ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) ) ) ).

% takeWhile_eq_all_conv
tff(fact_7436_takeWhile__append1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = takeWhile(A,P,Xs) ) ) ) ).

% takeWhile_append1
tff(fact_7437_takeWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) )
     => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys)) ) ) ).

% takeWhile_append2
tff(fact_7438_takeWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Nb: nat,Xb: A] :
      takeWhile(A,P,replicate(A,Nb,Xb)) = $ite(aa(A,$o,P,Xb),replicate(A,Nb,Xb),nil(A)) ).

% takeWhile_replicate
tff(fact_7439_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_7440_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_7441_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),F2: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xs))
       => distinct(B,Xs) ) ) ).

% folding_insort_key.distinct_if_distinct_map
tff(fact_7442_takeWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xs: list(B)] : takeWhile(A,P,aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),takeWhile(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P),F2),Xs)) ).

% takeWhile_map
tff(fact_7443_takeWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      takeWhile(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P,Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),takeWhile(A,P,Xs)),nil(A)) ).

% takeWhile.simps(2)
tff(fact_7444_takeWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : takeWhile(A,P,nil(A)) = nil(A) ).

% takeWhile.simps(1)
tff(fact_7445_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),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => inj_on(B,A,F2,S2) ) ).

% folding_insort_key.inj_on
tff(fact_7446_takeWhile__tail,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A),L: list(A)] :
      ( ~ aa(A,$o,P,Xb)
     => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),L))) = takeWhile(A,P,Xs) ) ) ).

% takeWhile_tail
tff(fact_7447_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_takeWhile_le
tff(fact_7448_takeWhile__eq__take,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : takeWhile(A,P,Xs) = take(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)),Xs) ).

% takeWhile_eq_take
tff(fact_7449_set__takeWhileD,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),takeWhile(A,P,Xs)))
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
        & aa(A,$o,P,Xb) ) ) ).

% set_takeWhileD
tff(fact_7450_takeWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P,X3)
            <=> aa(A,$o,Q,X3) ) )
       => ( takeWhile(A,P,L) = takeWhile(A,Q,K) ) ) ) ).

% takeWhile_cong
tff(fact_7451_takeWhile__eq__Nil__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( takeWhile(A,P,Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ~ aa(A,$o,P,aa(list(A),A,hd(A),Xs)) ) ) ).

% takeWhile_eq_Nil_iff
tff(fact_7452_distinct__takeWhile,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,takeWhile(A,P,Xs)) ) ).

% distinct_takeWhile
tff(fact_7453_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,takeWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_7454_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))
     => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ).

% nth_length_takeWhile
tff(fact_7455_takeWhile__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] :
      takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(
        ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) ),
        aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys)),
        takeWhile(A,P,Xs) ) ).

% takeWhile_append
tff(fact_7456_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_7457_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
         => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) )
       => ( takeWhile(A,P,Xs) = take(A,Nb,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_7458_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_xr(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_7459_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Ta: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F2),Xs)))
         => ( aa(list(B),list(B),filter2(B,aa(A,fun(B,$o),aTP_Lamp_ago(fun(B,A),fun(A,fun(B,$o)),F2),Ta)),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_ago(fun(B,A),fun(A,fun(B,$o)),F2),Ta),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_7460_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,$o)),S2: set(B),F2: fun(B,A),A3: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => ( aa(set(B),$o,finite_finite2(B),A3)
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),L))
              & ( aa(list(B),set(B),set2(B),L) = A3 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A3) ) )
          <=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_7461_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,$o)),S2: set(B),F2: fun(B,A),Xb: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xb),A3)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A3)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),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),F2),A3)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_7462_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_7463_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,$o)),S2: set(B),F2: fun(B,A),A3: set(B),B3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),S2)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),B3) )
           => ( aa(set(B),$o,finite_finite2(B),A3)
             => ( aa(set(B),$o,finite_finite2(B),B3)
               => ( A3 = B3 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_7464_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),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_7465_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,$o)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => ( aa(set(B),$o,finite_finite2(B),A3)
         => ( aa(list(B),set(B),set2(B),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) = A3 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_7466_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,$o)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => ( aa(list(B),nat,size_size(list(B)),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) = aa(set(B),nat,finite_card(B),A3) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_7467_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,$o)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => distinct(A,aa(list(B),list(A),map(B,A,F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_7468_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,$o)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_7469_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,$o)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_7470_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,$o)),S2: set(B),F2: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),S2)
       => ( aa(set(B),$o,finite_finite2(B),A3)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3) = nil(B) )
          <=> ( A3 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7471_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F2: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S2)
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F2),Xs))
         => ( distinct(B,Xs)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_7472_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,$o)),S2: set(B),F2: fun(B,A),Xb: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xb),A3)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A3)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xb),A3)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Xb),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xb),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_7473_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,$o)),S2: set(B),F2: fun(B,A),Xb: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xb),A3)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A3)
         => ( ~ member(B,Xb,A3)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xb),A3)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Xb),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A3)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_7474_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_7475_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),F2: fun(B,A),Xb: B,Y: B] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F2)
     => ( member(B,Xb,S2)
       => ( member(B,Y,S2)
         => ( aa(fun(list(B),list(B)),fun(list(B),list(B)),comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),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),F2),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),F2),Y)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_7476_extract__def,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : extract(A,P,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_agp(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),dropWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P),Xs)) ).

% extract_def
tff(fact_7477_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_7478_dropWhile__idem,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : dropWhile(A,P,dropWhile(A,P,Xs)) = dropWhile(A,P,Xs) ).

% dropWhile_idem
tff(fact_7479_dropWhile__eq__Nil__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) = nil(A) )
    <=> ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) ) ) ).

% dropWhile_eq_Nil_conv
tff(fact_7480_dropWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) )
     => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = dropWhile(A,P,Ys) ) ) ).

% dropWhile_append2
tff(fact_7481_dropWhile__append1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys) ) ) ) ).

% dropWhile_append1
tff(fact_7482_dropWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Nb: nat,Xb: A] :
      dropWhile(A,P,replicate(A,Nb,Xb)) = $ite(aa(A,$o,P,Xb),nil(A),replicate(A,Nb,Xb)) ).

% dropWhile_replicate
tff(fact_7483_takeWhile__dropWhile__id,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),takeWhile(A,P,Xs)),dropWhile(A,P,Xs)) = Xs ).

% takeWhile_dropWhile_id
tff(fact_7484_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_7485_distinct__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,dropWhile(A,P,Xs)) ) ).

% distinct_dropWhile
tff(fact_7486_hd__dropWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) != nil(A) )
     => ~ aa(A,$o,P,aa(list(A),A,hd(A),dropWhile(A,P,Xs))) ) ).

% hd_dropWhile
tff(fact_7487_dropWhile__eq__self__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) = Xs )
    <=> ( ( Xs = nil(A) )
        | ~ aa(A,$o,P,aa(list(A),A,hd(A),Xs)) ) ) ).

% dropWhile_eq_self_iff
tff(fact_7488_set__dropWhileD,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),dropWhile(A,P,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_dropWhileD
tff(fact_7489_dropWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K )
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P,X3)
            <=> aa(A,$o,Q,X3) ) )
       => ( dropWhile(A,P,L) = dropWhile(A,Q,K) ) ) ) ).

% dropWhile_cong
tff(fact_7490_length__dropWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_dropWhile_le
tff(fact_7491_dropWhile__append3,axiom,
    ! [A: $tType,P: fun(A,$o),Y: A,Xs: list(A),Ys: list(A)] :
      ( ~ aa(A,$o,P,Y)
     => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ).

% dropWhile_append3
tff(fact_7492_remdups__adj__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),remdups_adj(A,dropWhile(A,aTP_Lamp_bm(A,fun(A,$o),Xb),Xs))) ).

% remdups_adj_Cons'
tff(fact_7493_dropWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      dropWhile(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P,Xb),dropWhile(A,P,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) ).

% dropWhile.simps(2)
tff(fact_7494_dropWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : dropWhile(A,P,nil(A)) = nil(A) ).

% dropWhile.simps(1)
tff(fact_7495_dropWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),Xs: list(B)] : dropWhile(A,P,aa(list(B),list(A),map(B,A,F2),Xs)) = aa(list(B),list(A),map(B,A,F2),dropWhile(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P),F2),Xs)) ).

% dropWhile_map
tff(fact_7496_dropWhile__eq__Cons__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Y: A,Ys: list(A)] :
      ( ( dropWhile(A,P,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),takeWhile(A,P,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) )
        & ~ aa(A,$o,P,Y) ) ) ).

% dropWhile_eq_Cons_conv
tff(fact_7497_takeWhile__eq__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),dropWhile(A,P,Xs)))
         => ~ aa(A,$o,P,X3) )
     => ( takeWhile(A,P,Xs) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% takeWhile_eq_filter
tff(fact_7498_dropWhile__eq__drop,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : dropWhile(A,P,Xs) = drop(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)),Xs) ).

% dropWhile_eq_drop
tff(fact_7499_dropWhile__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] :
      dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(
        ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) ),
        dropWhile(A,P,Ys),
        aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys) ) ).

% dropWhile_append
tff(fact_7500_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : find(A,P,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_agq(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P),Xs)) ).

% find_dropWhile
tff(fact_7501_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),remdups_adj(A,dropWhile(A,aTP_Lamp_bm(A,fun(A,$o),Y),Ys))) ).

% remdups_adj_append_dropWhile
tff(fact_7502_tl__remdups__adj,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( ( Ys != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys)) = remdups_adj(A,dropWhile(A,aTP_Lamp_agr(list(A),fun(A,$o),Ys),aa(list(A),list(A),tl(A),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_7503_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,dropWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_7504_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_7505_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( dropWhile(A,aa(A,fun(A,$o),aTP_Lamp_afm(A,fun(A,$o)),Xb),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),rev(A),takeWhile(A,aa(A,fun(A,$o),aTP_Lamp_afm(A,fun(A,$o)),Xb),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_7506_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( takeWhile(A,aa(A,fun(A,$o),aTP_Lamp_afm(A,fun(A,$o)),Xb),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),dropWhile(A,aa(A,fun(A,$o),aTP_Lamp_afm(A,fun(A,$o)),Xb),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_7507_partition__filter__conv,axiom,
    ! [A: $tType,F2: fun(A,$o),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F2),Xs) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),filter2(A,F2),Xs)),aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),F2)),Xs)) ).

% partition_filter_conv
tff(fact_7508_max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : max_ext(A,R) = collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),$o,aTP_Lamp_ags(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R))) ).

% max_ext_eq
tff(fact_7509_bex__empty,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ~ ? [X: A] :
          ( member(A,X,bot_bot(set(A)))
          & aa(A,$o,P,X) ) ).

% bex_empty
tff(fact_7510_finite__Collect__bex,axiom,
    ! [B: $tType,A: $tType,A3: set(A),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(B),$o,finite_finite2(B),collect(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_agt(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),Q)))
      <=> ! [X2: A] :
            ( member(A,X2,A3)
           => aa(set(B),$o,finite_finite2(B),collect(B,aa(A,fun(B,$o),aTP_Lamp_ys(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),X2))) ) ) ) ).

% finite_Collect_bex
tff(fact_7511_bex__UNIV,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X2: A] :
          ( member(A,X2,top_top(set(A)))
          & aa(A,$o,P,X2) )
    <=> ? [X_13: A] : aa(A,$o,P,X_13) ) ).

% bex_UNIV
tff(fact_7512_image__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(set(B),set(A),image2(B,A,F2),A3) = collect(A,aa(set(B),fun(A,$o),aTP_Lamp_agu(fun(B,A),fun(set(B),fun(A,$o)),F2),A3)) ).

% image_def
tff(fact_7513_Bex__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ? [X2: A] :
            ( member(A,X2,A3)
            & aa(A,$o,P,X2) )
      <=> finite_fold(A,$o,aTP_Lamp_agv(fun(A,$o),fun(A,fun($o,$o)),P),$false,A3) ) ) ).

% Bex_fold
tff(fact_7514_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B3: set(nat)] : nths(A,nths(A,Xs,A3),B3) = nths(A,Xs,collect(nat,aa(set(nat),fun(nat,$o),aTP_Lamp_agx(set(nat),fun(set(nat),fun(nat,$o)),A3),B3))) ).

% nths_nths
tff(fact_7515_partition_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),nil(A)) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),nil(A)) ).

% partition.simps(1)
tff(fact_7516_partition__P,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Yes),No4) )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Yes))
           => aa(A,$o,P,X) )
        & ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),No4))
           => ~ aa(A,$o,P,X) ) ) ) ).

% partition_P
tff(fact_7517_partition_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(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_agy(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P),Xb)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) ).

% partition.simps(2)
tff(fact_7518_partition__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),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_7519_min__ext__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : min_ext(A,R2) = collect(product_prod(set(A),set(A)),aTP_Lamp_agz(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R2)) ).

% min_ext_def
tff(fact_7520_map__project__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),A3: set(B)] : map_project(B,A,F2,A3) = collect(A,aa(set(B),fun(A,$o),aTP_Lamp_aha(fun(B,option(A)),fun(set(B),fun(A,$o)),F2),A3)) ).

% map_project_def
tff(fact_7521_max__extp_Ocases,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
      ( max_extp(A,R,A1,A22)
     => ~ ( aa(set(A),$o,finite_finite2(A),A1)
         => ( aa(set(A),$o,finite_finite2(A),A22)
           => ( ( A22 != collect(A,bot_bot(fun(A,$o))) )
             => ~ ! [X: A] :
                    ( member(A,X,A1)
                   => ? [Xa2: A] :
                        ( member(A,Xa2,A22)
                        & aa(A,$o,aa(A,fun(A,$o),R,X),Xa2) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_7522_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 != 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) ) )
           => max_extp(A,R,X5,Y5) ) ) ) ) ).

% max_extp.max_extI
tff(fact_7523_max__extp_Osimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),A1: set(A),A22: set(A)] :
      ( max_extp(A,R,A1,A22)
    <=> ( aa(set(A),$o,finite_finite2(A),A1)
        & aa(set(A),$o,finite_finite2(A),A22)
        & ( A22 != collect(A,bot_bot(fun(A,$o))) )
        & ! [X2: A] :
            ( member(A,X2,A1)
           => ? [Xa3: A] :
                ( member(A,Xa3,A22)
                & aa(A,$o,aa(A,fun(A,$o),R,X2),Xa3) ) ) ) ) ).

% max_extp.simps
tff(fact_7524_prod_Oset__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(list(B),set(B),set2(B),Xs)) = groups5270119922927024881d_list(A,aa(list(B),list(A),map(B,A,G),remdups(B,Xs))) ) ).

% prod.set_conv_list
tff(fact_7525_lists__empty,axiom,
    ! [A: $tType] : lists(A,bot_bot(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% lists_empty
tff(fact_7526_Cons__in__lists__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),A3: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),lists(A,A3))
    <=> ( member(A,Xb,A3)
        & member(list(A),Xs,lists(A,A3)) ) ) ).

% Cons_in_lists_iff
tff(fact_7527_in__listsI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => member(A,X3,A3) )
     => member(list(A),Xs,lists(A,A3)) ) ).

% in_listsI
tff(fact_7528_lists__Int__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,A3)),lists(A,B3)) ).

% lists_Int_eq
tff(fact_7529_append__in__lists__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),A3: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),lists(A,A3))
    <=> ( member(list(A),Xs,lists(A,A3))
        & member(list(A),Ys,lists(A,A3)) ) ) ).

% append_in_lists_conv
tff(fact_7530_lists__UNIV,axiom,
    ! [A: $tType] : lists(A,top_top(set(A))) = top_top(set(list(A))) ).

% lists_UNIV
tff(fact_7531_lists__IntI,axiom,
    ! [A: $tType,L: list(A),A3: set(A),B3: set(A)] :
      ( member(list(A),L,lists(A,A3))
     => ( member(list(A),L,lists(A,B3))
       => member(list(A),L,lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ).

% lists_IntI
tff(fact_7532_lists_Osimps,axiom,
    ! [A: $tType,A2: list(A),A3: set(A)] :
      ( member(list(A),A2,lists(A,A3))
    <=> ( ( A2 = nil(A) )
        | ? [A7: A,L4: list(A)] :
            ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),L4) )
            & member(A,A7,A3)
            & member(list(A),L4,lists(A,A3)) ) ) ) ).

% lists.simps
tff(fact_7533_lists_Ocases,axiom,
    ! [A: $tType,A2: list(A),A3: set(A)] :
      ( member(list(A),A2,lists(A,A3))
     => ( ( A2 != nil(A) )
       => ~ ! [A4: A,L3: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),L3) )
             => ( member(A,A4,A3)
               => ~ member(list(A),L3,lists(A,A3)) ) ) ) ) ).

% lists.cases
tff(fact_7534_lists_ONil,axiom,
    ! [A: $tType,A3: set(A)] : member(list(A),nil(A),lists(A,A3)) ).

% lists.Nil
tff(fact_7535_listrel__refl__on,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A3,R2)
     => refl_on(list(A),lists(A,A3),listrel(A,A,R2)) ) ).

% listrel_refl_on
tff(fact_7536_listsE,axiom,
    ! [A: $tType,Xb: A,L: list(A),A3: set(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),L),lists(A,A3))
     => ~ ( member(A,Xb,A3)
         => ~ member(list(A),L,lists(A,A3)) ) ) ).

% listsE
tff(fact_7537_lists_OCons,axiom,
    ! [A: $tType,A2: A,A3: set(A),L: list(A)] :
      ( member(A,A2,A3)
     => ( member(list(A),L,lists(A,A3))
       => member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),L),lists(A,A3)) ) ) ).

% lists.Cons
tff(fact_7538_lists__eq__set,axiom,
    ! [A: $tType,A3: set(A)] : lists(A,A3) = collect(list(A),aTP_Lamp_ahb(set(A),fun(list(A),$o),A3)) ).

% lists_eq_set
tff(fact_7539_in__listsD,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( member(list(A),Xs,lists(A,A3))
     => ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => member(A,X,A3) ) ) ).

% in_listsD
tff(fact_7540_in__lists__conv__set,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( member(list(A),Xs,lists(A,A3))
    <=> ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => member(A,X2,A3) ) ) ).

% in_lists_conv_set
tff(fact_7541_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_7542_lists__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),lists(A,A3)),lists(A,B3)) ) ).

% lists_mono
tff(fact_7543_lists__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : lists(A,aa(set(B),set(A),image2(B,A,F2),A3)) = aa(set(list(B)),set(list(A)),image2(list(B),list(A),map(B,A,F2)),lists(B,A3)) ).

% lists_image
tff(fact_7544_Collect__finite__eq__lists,axiom,
    ! [A: $tType] : 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_7545_Collect__finite__subset__eq__lists,axiom,
    ! [A: $tType,T3: set(A)] : collect(set(A),aTP_Lamp_abg(set(A),fun(set(A),$o),T3)) = aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),lists(A,T3)) ).

% Collect_finite_subset_eq_lists
tff(fact_7546_listrel__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_adl(set(A),fun(A,set(A)),A3)))
     => aa(set(product_prod(list(A),list(A))),$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,A3),aTP_Lamp_ahc(set(A),fun(list(A),set(list(A))),A3))) ) ).

% listrel_subset
tff(fact_7547_prod_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Xs: list(A),G: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(list(A),set(A),set2(A),Xs)) = groups5270119922927024881d_list(B,aa(list(A),list(B),map(A,B,G),Xs)) ) ) ) ).

% prod.distinct_set_conv_list
tff(fact_7548_card__quotient__disjoint,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( inj_on(A,set(set(A)),aTP_Lamp_ahd(set(product_prod(A,A)),fun(A,set(set(A))),R2),A3)
       => ( aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A3,R2)) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_quotient_disjoint
tff(fact_7549_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,aa(list(B),fun(A,A),fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,inf_inf(A)),F2)),Xs),top_top(A)) ) ).

% INF_set_fold
tff(fact_7550_fold__append,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xs: list(B),Ys: list(B)] : aa(list(B),fun(A,A),fold(B,A,F2),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(list(B),fun(A,A),fold(B,A,F2),Ys)),aa(list(B),fun(A,A),fold(B,A,F2),Xs)) ).

% fold_append
tff(fact_7551_fold__replicate,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Nb: nat,Xb: B] : aa(list(B),fun(A,A),fold(B,A,F2),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),F2,Xb)) ).

% fold_replicate
tff(fact_7552_foldr__conv__fold,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xs: list(B)] : foldr(B,A,F2,Xs) = aa(list(B),fun(A,A),fold(B,A,F2),aa(list(B),list(B),rev(B),Xs)) ).

% foldr_conv_fold
tff(fact_7553_fold__invariant,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Q: fun(A,$o),P: fun(B,$o),S: B,F2: fun(A,fun(B,B))] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,Q,X3) )
     => ( aa(B,$o,P,S)
       => ( ! [X3: A,S3: B] :
              ( aa(A,$o,Q,X3)
             => ( aa(B,$o,P,S3)
               => aa(B,$o,P,aa(B,B,aa(A,fun(B,B),F2,X3),S3)) ) )
         => aa(B,$o,P,aa(B,B,aa(list(A),fun(B,B),fold(A,B,F2),Xs),S)) ) ) ) ).

% fold_invariant
tff(fact_7554_List_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,Xs: list(B),Ys: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X3: B] :
              ( member(B,X3,aa(list(B),set(B),set2(B),Xs))
             => ( aa(B,fun(A,A),F2,X3) = aa(B,fun(A,A),G,X3) ) )
         => ( aa(A,A,aa(list(B),fun(A,A),fold(B,A,F2),Xs),A2) = aa(A,A,aa(list(B),fun(A,A),fold(B,A,G),Ys),B2) ) ) ) ) ).

% List.fold_cong
tff(fact_7555_fold__id,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,fun(B,B),F2,X3) = id(B) ) )
     => ( aa(list(A),fun(B,B),fold(A,B,F2),Xs) = id(B) ) ) ).

% fold_id
tff(fact_7556_fold__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Xb: B,Xs: list(B),S: A] : aa(A,A,aa(list(B),fun(A,A),fold(B,A,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)),S) = aa(A,A,aa(list(B),fun(A,A),fold(B,A,F2),Xs),aa(A,A,aa(B,fun(A,A),F2,Xb),S)) ).

% fold_simps(2)
tff(fact_7557_fold__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),S: A] : aa(A,A,aa(list(B),fun(A,A),fold(B,A,F2),nil(B)),S) = S ).

% fold_simps(1)
tff(fact_7558_fold__Nil,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A))] : aa(list(B),fun(A,A),fold(B,A,F2),nil(B)) = id(A) ).

% fold_Nil
tff(fact_7559_fold__filter,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),P: fun(B,$o),Xs: list(B)] : aa(list(B),fun(A,A),fold(B,A,F2),aa(list(B),list(B),filter2(B,P),Xs)) = aa(list(B),fun(A,A),fold(B,A,aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_afz(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F2),P)),Xs) ).

% fold_filter
tff(fact_7560_fold__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,fun(A,A)),F2: fun(C,B),Xs: list(C)] : aa(list(B),fun(A,A),fold(B,A,G),aa(list(C),list(B),map(C,B,F2),Xs)) = aa(list(C),fun(A,A),fold(C,A,aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G),F2)),Xs) ).

% fold_map
tff(fact_7561_fold__commute,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F2: fun(A,fun(C,C))] :
      ( ! [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),G,X3)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F2,X3)),H) ) )
     => ( aa(fun(B,B),fun(B,C),comp(B,C,B,H),aa(list(A),fun(B,B),fold(A,B,G),Xs)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(list(A),fun(C,C),fold(A,C,F2),Xs)),H) ) ) ).

% fold_commute
tff(fact_7562_fold__commute__apply,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F2: fun(A,fun(C,C)),S: B] :
      ( ! [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),G,X3)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F2,X3)),H) ) )
     => ( aa(B,C,H,aa(B,B,aa(list(A),fun(B,B),fold(A,B,G),Xs),S)) = aa(C,C,aa(list(A),fun(C,C),fold(A,C,F2),Xs),aa(B,C,H,S)) ) ) ).

% fold_commute_apply
tff(fact_7563_fold__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),Xb: B,Xs: list(B)] : aa(list(B),fun(A,A),fold(B,A,F2),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(list(B),fun(A,A),fold(B,A,F2),Xs)),aa(B,fun(A,A),F2,Xb)) ).

% fold_Cons
tff(fact_7564_union__set__fold,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),A3) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),fold(A,set(A),insert2(A)),Xs),A3) ).

% union_set_fold
tff(fact_7565_rev__conv__fold,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),fold(A,list(A),cons(A)),Xs),nil(A)) ).

% rev_conv_fold
tff(fact_7566_fold__Cons__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),fun(list(A),list(A)),fold(A,list(A),cons(A)),Xs) = aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xs)) ).

% fold_Cons_rev
tff(fact_7567_fold__rev,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y3)) ) ) )
     => ( aa(list(A),fun(B,B),fold(A,B,F2),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),fun(B,B),fold(A,B,F2),Xs) ) ) ).

% fold_rev
tff(fact_7568_fold__plus__sum__list__rev,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : aa(list(A),fun(A,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_7569_fold__remove1__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B)),Xb: A] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X3)) ) ) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( aa(list(A),fun(B,B),fold(A,B,F2),Xs) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(list(A),fun(B,B),fold(A,B,F2),remove1(A,Xb,Xs))),aa(A,fun(B,B),F2,Xb)) ) ) ) ).

% fold_remove1_split
tff(fact_7570_foldr__fold,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y3)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y3)) ) ) )
     => ( foldr(A,B,F2,Xs) = aa(list(A),fun(B,B),fold(A,B,F2),Xs) ) ) ).

% foldr_fold
tff(fact_7571_minus__set__fold,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(list(A),set(A),set2(A),Xs)) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),fold(A,set(A),remove(A)),Xs),A3) ).

% minus_set_fold
tff(fact_7572_fold__append__concat__rev,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),fun(list(A),list(A)),fold(list(A),list(A),append(A)),Xss) = aa(list(A),fun(list(A),list(A)),append(A),concat(A,aa(list(list(A)),list(list(A)),rev(list(A)),Xss))) ).

% fold_append_concat_rev
tff(fact_7573_Sup__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,aa(list(A),fun(A,A),fold(A,A,sup_sup(A)),Xs),bot_bot(A)) ) ).

% Sup_set_fold
tff(fact_7574_Inf__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,aa(list(A),fun(A,A),fold(A,A,inf_inf(A)),Xs),top_top(A)) ) ).

% Inf_set_fold
tff(fact_7575_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,aa(list(A),fun(A,A),fold(A,A,inf_inf(A)),Xs),Xb) ) ).

% Inf_fin.set_eq_fold
tff(fact_7576_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,aa(list(A),fun(A,A),fold(A,A,sup_sup(A)),Xs),Xb) ) ).

% Sup_fin.set_eq_fold
tff(fact_7577_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,aa(list(A),fun(A,A),fold(A,A,ord_max(A)),Xs),Xb) ) ).

% Max.set_eq_fold
tff(fact_7578_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,aa(list(A),fun(A,A),fold(A,A,ord_min(A)),Xs),Xb) ) ).

% Min.set_eq_fold
tff(fact_7579_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => ( aa(set(A),$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,F2,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,aa(list(A),fun(B,B),fold(A,B,F2),Xs),Y) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_7580_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$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,F2,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,aa(list(A),fun(B,B),fold(A,B,F2),remdups(A,Xs)),Y) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_7581_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,aa(list(B),fun(A,A),fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F2)),Xs),bot_bot(A)) ) ).

% SUP_set_fold
tff(fact_7582_quotient__is__empty2,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( ( bot_bot(set(set(A))) = equiv_quotient(A,A3,R2) )
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% quotient_is_empty2
tff(fact_7583_quotient__is__empty,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( ( equiv_quotient(A,A3,R2) = bot_bot(set(set(A))) )
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% quotient_is_empty
tff(fact_7584_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_7585_finite__equiv__class,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( 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,A3,aTP_Lamp_adl(set(A),fun(A,set(A)),A3)))
       => ( member(set(A),X5,equiv_quotient(A,A3,R2))
         => aa(set(A),$o,finite_finite2(A),X5) ) ) ) ).

% finite_equiv_class
tff(fact_7586_finite__quotient,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( 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,A3,aTP_Lamp_adl(set(A),fun(A,set(A)),A3)))
       => aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A3,R2)) ) ) ).

% finite_quotient
tff(fact_7587_quotient__diff1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),A2: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_ahd(set(product_prod(A,A)),fun(A,set(set(A))),R2),A3)
     => ( member(A,A2,A3)
       => ( equiv_quotient(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))),R2) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),minus_minus(set(set(A))),equiv_quotient(A,A3,R2)),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))),R2)) ) ) ) ).

% quotient_diff1
tff(fact_7588_sort__key__conv__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] :
          ( inj_on(A,B,F2,aa(list(A),set(A),set2(A),Xs))
         => ( aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),fold(A,list(A),linorder_insort_key(A,B,F2)),Xs),nil(A)) ) ) ) ).

% sort_key_conv_fold
tff(fact_7589_inter__coset__fold,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),coset(A,Xs)) = aa(set(A),set(A),aa(list(A),fun(set(A),set(A)),fold(A,set(A),remove(A)),Xs),A3) ).

% inter_coset_fold
tff(fact_7590_sort__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa(list(nat),list(nat),aa(fun(nat,nat),fun(list(nat),list(nat)),linorder_sort_key(nat,nat),aTP_Lamp_bk(nat,nat)),upt(Mb,Nb)) = upt(Mb,Nb) ).

% sort_upt
tff(fact_7591_sort__upto,axiom,
    ! [Ia: int,J: int] : aa(list(int),list(int),aa(fun(int,int),fun(list(int),list(int)),linorder_sort_key(int,int),aTP_Lamp_bj(int,int)),upto(Ia,J)) = upto(Ia,J) ).

% sort_upto
tff(fact_7592_sort__key__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B)] : aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),nil(A)) = nil(A) ) ).

% sort_key_simps(1)
tff(fact_7593_set__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_sort
tff(fact_7594_length__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ).

% length_sort
tff(fact_7595_distinct__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] :
          ( distinct(A,aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs))
        <=> distinct(A,Xs) ) ) ).

% distinct_sort
tff(fact_7596_sort__key__simps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),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,F2),Xb),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs)) ) ).

% sort_key_simps(2)
tff(fact_7597_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),aTP_Lamp_qf(B,fun(A,B),C2)),Xs) = Xs ) ).

% sort_key_const
tff(fact_7598_filter__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F2: fun(A,B),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs)) = aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% filter_sort
tff(fact_7599_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,$o)),F2),K)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,$o)),F2),K)),Xs) ) ).

% sort_key_stable
tff(fact_7600_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_xr(A,A)),Xs)) ) ).

% sorted_sort
tff(fact_7601_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_xr(A,A)),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_7602_UNIV__coset,axiom,
    ! [A: $tType] : top_top(set(A)) = coset(A,nil(A)) ).

% UNIV_coset
tff(fact_7603_subset__code_I2_J,axiom,
    ! [A: $tType,A3: set(A),Ys: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),coset(A,Ys))
    <=> ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Ys))
         => ~ member(A,X2,A3) ) ) ).

% subset_code(2)
tff(fact_7604_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_7605_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_7606_insert__code_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),coset(A,Xs)) = coset(A,aa(list(A),list(A),removeAll(A,Xb),Xs)) ).

% insert_code(2)
tff(fact_7607_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),linorder_sort_key(B,A),F2),Xs))) ) ).

% sorted_sort_key
tff(fact_7608_union__coset__filter,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xs)),A3) = coset(A,aa(list(A),list(A),filter2(A,aTP_Lamp_bc(set(A),fun(A,$o),A3)),Xs)) ).

% union_coset_filter
tff(fact_7609_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_7610_sorted__list__of__set__sort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_xr(A,A)),remdups(A,Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_7611_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_xr(A,A)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_xr(A,A))),Xs),nil(A)) ) ).

% sort_conv_fold
tff(fact_7612_minus__coset__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),coset(A,Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_a(set(A),fun(A,$o),A3)),Xs)) ).

% minus_coset_filter
tff(fact_7613_sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A)] : aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),linorder_sort_key(A,B),F2),Xs) = aa(list(A),list(A),foldr(A,list(A),linorder_insort_key(A,B,F2),Xs),nil(A)) ) ).

% sort_key_def
tff(fact_7614_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P) = aa(list(A),A,case_list(A,A,abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P),aTP_Lamp_ahf(A,fun(list(A),A))),aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),linorder_sort_key(A,A),aTP_Lamp_xr(A,A)),Xs))) ) ).

% Bleast_code
tff(fact_7615_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X5: set(A)] :
      ( countable_countable(A,X5)
     => ~ ! [F5: fun(nat,set(A))] :
            ( ! [I3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F5,I3)),X5)
           => ( ! [I3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F5,I3)),aa(nat,set(A),F5,aa(nat,nat,suc,I3)))
             => ( ! [I3: nat] : aa(set(A),$o,finite_finite2(A),aa(nat,set(A),F5,I3))
               => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),F5),top_top(set(nat)))) != X5 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_7616_countable__empty,axiom,
    ! [A: $tType] : countable_countable(A,bot_bot(set(A))) ).

% countable_empty
tff(fact_7617_countable__Diff__eq,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))))
    <=> countable_countable(A,A3) ) ).

% countable_Diff_eq
tff(fact_7618_countable__image__eq__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image2(B,A,F2),S2))
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S2)
          & ( aa(set(B),set(A),image2(B,A,F2),S2) = aa(set(B),set(A),image2(B,A,F2),T9) )
          & inj_on(B,A,F2,T9) ) ) ).

% countable_image_eq_inj
tff(fact_7619_ex__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),$o)] :
      ( ? [T9: set(A)] :
          ( countable_countable(A,T9)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S2))
          & aa(set(A),$o,P,T9) )
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S2)
          & inj_on(B,A,F2,T9)
          & aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% ex_countable_subset_image_inj
tff(fact_7620_all__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),$o)] :
      ( ! [T9: set(A)] :
          ( ( countable_countable(A,T9)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S2)) )
         => aa(set(A),$o,P,T9) )
    <=> ! [T9: set(B)] :
          ( ( countable_countable(B,T9)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S2)
            & inj_on(B,A,F2,T9) )
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% all_countable_subset_image_inj
tff(fact_7621_ccSup__subset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ) ).

% ccSup_subset_mono
tff(fact_7622_to__nat__on__surj,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( countable_countable(A,A3)
     => ( ~ aa(set(A),$o,finite_finite2(A),A3)
       => ? [X3: A] :
            ( member(A,X3,A3)
            & ( aa(A,nat,countable_to_nat_on(A,A3),X3) = Nb ) ) ) ) ).

% to_nat_on_surj
tff(fact_7623_countable__Collect__finite,axiom,
    ! [A: $tType] :
      ( countable(A)
     => countable_countable(set(A),collect(set(A),finite_finite2(A))) ) ).

% countable_Collect_finite
tff(fact_7624_uncountable__infinite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ countable_countable(A,A3)
     => ~ aa(set(A),$o,finite_finite2(A),A3) ) ).

% uncountable_infinite
tff(fact_7625_countable__finite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => countable_countable(A,S2) ) ).

% countable_finite
tff(fact_7626_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,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2)
          <=> ? [X2: A] :
                ( member(A,X2,S2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),A2) ) ) ) ) ).

% ccInf_less_iff
tff(fact_7627_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,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2))
          <=> ? [X2: A] :
                ( member(A,X2,S2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X2) ) ) ) ) ).

% less_ccSup_iff
tff(fact_7628_ccSup__upper2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),U: A,V2: A] :
          ( countable_countable(A,A3)
         => ( member(A,U,A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),U)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ) ).

% ccSup_upper2
tff(fact_7629_ccSup__le__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B2: A] :
          ( countable_countable(A,A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2)
          <=> ! [X2: A] :
                ( member(A,X2,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B2) ) ) ) ) ).

% ccSup_le_iff
tff(fact_7630_ccSup__upper,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Xb: A] :
          ( countable_countable(A,A3)
         => ( member(A,Xb,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).

% ccSup_upper
tff(fact_7631_ccSup__least,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Z: A] :
          ( countable_countable(A,A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(A,$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),A3)),Z) ) ) ) ).

% ccSup_least
tff(fact_7632_ccSup__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( countable_countable(A,A3)
           => ( ! [A4: A] :
                  ( member(A,A4,A3)
                 => ? [X: A] :
                      ( member(A,X,B3)
                      & aa(A,$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),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)) ) ) ) ) ).

% ccSup_mono
tff(fact_7633_countable__image__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image2(B,A,F2),S2))
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S2)
          & ( aa(set(B),set(A),image2(B,A,F2),S2) = aa(set(B),set(A),image2(B,A,F2),T9) ) ) ) ).

% countable_image_eq
tff(fact_7634_countable__subset__image,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F2: fun(B,A),A3: set(B)] :
      ( ( countable_countable(A,B3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3)) )
    <=> ? [A16: set(B)] :
          ( countable_countable(B,A16)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A16),A3)
          & ( B3 = aa(set(B),set(A),image2(B,A,F2),A16) ) ) ) ).

% countable_subset_image
tff(fact_7635_ex__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),$o)] :
      ( ? [T9: set(A)] :
          ( countable_countable(A,T9)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S2))
          & aa(set(A),$o,P,T9) )
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S2)
          & aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% ex_countable_subset_image
tff(fact_7636_all__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S2: set(B),P: fun(set(A),$o)] :
      ( ! [T9: set(A)] :
          ( ( countable_countable(A,T9)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S2)) )
         => aa(set(A),$o,P,T9) )
    <=> ! [T9: set(B)] :
          ( ( countable_countable(B,T9)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S2) )
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% all_countable_subset_image
tff(fact_7637_countable__Collect__finite__subset,axiom,
    ! [A: $tType,T3: set(A)] :
      ( countable_countable(A,T3)
     => countable_countable(set(A),collect(set(A),aTP_Lamp_abg(set(A),fun(set(A),$o),T3))) ) ).

% countable_Collect_finite_subset
tff(fact_7638_infinite__countable__subset_H,axiom,
    ! [A: $tType,X5: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),X5)
     => ? [C8: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C8),X5)
          & countable_countable(A,C8)
          & ~ aa(set(A),$o,finite_finite2(A),C8) ) ) ).

% infinite_countable_subset'
tff(fact_7639_ccInf__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( countable_countable(A,A3)
           => ( ! [B4: A] :
                  ( member(A,B4,B3)
                 => ? [X: A] :
                      ( member(A,X,A3)
                      & aa(A,$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),A3)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ) ) ).

% ccInf_mono
tff(fact_7640_ccInf__lower,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Xb: A] :
          ( countable_countable(A,A3)
         => ( member(A,Xb,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),Xb) ) ) ) ).

% ccInf_lower
tff(fact_7641_ccInf__lower2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),U: A,V2: A] :
          ( countable_countable(A,A3)
         => ( member(A,U,A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V2) ) ) ) ) ).

% ccInf_lower2
tff(fact_7642_le__ccInf__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B2: A] :
          ( countable_countable(A,A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3))
          <=> ! [X2: A] :
                ( member(A,X2,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X2) ) ) ) ) ).

% le_ccInf_iff
tff(fact_7643_ccInf__greatest,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Z: A] :
          ( countable_countable(A,A3)
         => ( ! [X3: A] :
                ( member(A,X3,A3)
               => aa(A,$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),A3)) ) ) ) ).

% ccInf_greatest
tff(fact_7644_countable__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => ( countable_countable(A,B3)
       => countable_countable(A,A3) ) ) ).

% countable_subset
tff(fact_7645_ccInf__superset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3)) ) ) ) ).

% ccInf_superset_mono
tff(fact_7646_uncountable__def,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ countable_countable(A,A3)
    <=> ( ( A3 != bot_bot(set(A)) )
        & ~ ? [F7: fun(nat,A)] : aa(set(nat),set(A),image2(nat,A,F7),top_top(set(nat))) = A3 ) ) ).

% uncountable_def
tff(fact_7647_ccSUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),Ia: A,U: B,F2: fun(A,B)] :
          ( countable_countable(A,A3)
         => ( member(A,Ia,A3)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,Ia))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ) ).

% ccSUP_upper2
tff(fact_7648_ccSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( countable_countable(A,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),U)
          <=> ! [X2: A] :
                ( member(A,X2,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),U) ) ) ) ) ).

% ccSUP_le_iff
tff(fact_7649_ccSUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),Ia: A,F2: fun(A,B)] :
          ( countable_countable(A,A3)
         => ( member(A,Ia,A3)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ia)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ).

% ccSUP_upper
tff(fact_7650_ccSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( countable_countable(A,A3)
         => ( ! [I2: A] :
                ( member(A,I2,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),U) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),U) ) ) ) ).

% ccSUP_least
tff(fact_7651_ccSUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( counta3822494911875563373attice(C)
     => ! [A3: set(A),B3: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( countable_countable(A,A3)
         => ( countable_countable(B,B3)
           => ( ! [N: A] :
                  ( member(A,N,A3)
                 => ? [X: B] :
                      ( member(B,X,B3)
                      & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,N)),aa(B,C,G,X)) ) )
             => aa(C,$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,F2),A3))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B3))) ) ) ) ) ).

% ccSUP_mono
tff(fact_7652_countable__infiniteE_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => ( ~ aa(set(A),$o,finite_finite2(A),A3)
       => ~ ! [G6: fun(nat,A)] : ~ bij_betw(nat,A,G6,top_top(set(nat)),A3) ) ) ).

% countable_infiniteE'
tff(fact_7653_less__ccSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(B)
        & linorder(B) )
     => ! [A3: set(A),A2: B,F2: fun(A,B)] :
          ( countable_countable(A,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3)))
          <=> ? [X2: A] :
                ( member(A,X2,A3)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(A,B,F2,X2)) ) ) ) ) ).

% less_ccSUP_iff
tff(fact_7654_ccINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),U: B,F2: fun(A,B)] :
          ( countable_countable(A,A3)
         => ( ! [I2: A] :
                ( member(A,I2,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ).

% ccINF_greatest
tff(fact_7655_le__ccINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),U: B,F2: fun(A,B)] :
          ( countable_countable(A,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3)))
          <=> ! [X2: A] :
                ( member(A,X2,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,X2)) ) ) ) ) ).

% le_ccINF_iff
tff(fact_7656_ccINF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),Ia: A,F2: fun(A,B),U: B] :
          ( countable_countable(A,A3)
         => ( member(A,Ia,A3)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ia)),U)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),U) ) ) ) ) ).

% ccINF_lower2
tff(fact_7657_ccINF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),Ia: A,F2: fun(A,B)] :
          ( countable_countable(A,A3)
         => ( member(A,Ia,A3)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,Ia)) ) ) ) ).

% ccINF_lower
tff(fact_7658_ccINF__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( counta3822494911875563373attice(C)
     => ! [A3: set(A),B3: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( countable_countable(A,A3)
         => ( countable_countable(B,B3)
           => ( ! [M2: B] :
                  ( member(B,M2,B3)
                 => ? [X: A] :
                      ( member(A,X,A3)
                      & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X)),aa(B,C,G,M2)) ) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,F2),A3))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),B3))) ) ) ) ) ).

% ccINF_mono
tff(fact_7659_ccINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(B)
        & linorder(B) )
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( countable_countable(A,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),A2)
          <=> ? [X2: A] :
                ( member(A,X2,A3)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),A2) ) ) ) ) ).

% ccINF_less_iff
tff(fact_7660_countableE__infinite,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ~ aa(set(A),$o,finite_finite2(A),S2)
       => ~ ! [E: fun(A,nat)] : ~ bij_betw(A,nat,E,S2,top_top(set(nat))) ) ) ).

% countableE_infinite
tff(fact_7661_ccSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( countable_countable(A,B3)
           => aa(A,$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)),A3),B3))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ).

% ccSup_inter_less_eq
tff(fact_7662_less__eq__ccInf__inter,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( countable_countable(A,B3)
           => aa(A,$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),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% less_eq_ccInf_inter
tff(fact_7663_ccSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [B3: set(A),A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( countable_countable(A,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
           => ( ! [X3: A] :
                  ( member(A,X3,A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ) ).

% ccSUP_subset_mono
tff(fact_7664_ccINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),B3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( countable_countable(A,A3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
           => ( ! [X3: A] :
                  ( member(A,X3,B3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B3))) ) ) ) ) ).

% ccINF_superset_mono
tff(fact_7665_mono__ccSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( countable_countable(A,A3)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% mono_ccSup
tff(fact_7666_mono__ccSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),I5: set(C),A3: fun(C,A)] :
          ( order_mono(A,B,F2)
         => ( countable_countable(C,I5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahg(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A3),I5)))) ) ) ) ).

% mono_ccSUP
tff(fact_7667_mono__ccINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta4013691401010221786attice(A) )
     => ! [F2: fun(A,B),I5: set(C),A3: fun(C,A)] :
          ( order_mono(A,B,F2)
         => ( countable_countable(C,I5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahg(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))) ) ) ) ).

% mono_ccINF
tff(fact_7668_mono__ccInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( countable_countable(A,A3)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A3))) ) ) ) ).

% mono_ccInf
tff(fact_7669_countable__as__injective__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => ( ~ aa(set(A),$o,finite_finite2(A),A3)
       => ~ ! [F3: fun(nat,A)] :
              ( ( A3 = aa(set(nat),set(A),image2(nat,A,F3),top_top(set(nat))) )
             => ~ inj_on(nat,A,F3,top_top(set(nat))) ) ) ) ).

% countable_as_injective_image
tff(fact_7670_image__to__nat__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => ( ~ aa(set(A),$o,finite_finite2(A),A3)
       => ( aa(set(A),set(nat),image2(A,nat,countable_to_nat_on(A,A3)),A3) = top_top(set(nat)) ) ) ) ).

% image_to_nat_on
tff(fact_7671_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_7672_countable__enum__cases,axiom,
    ! [A: $tType,S2: set(A)] :
      ( countable_countable(A,S2)
     => ( ( aa(set(A),$o,finite_finite2(A),S2)
         => ! [F3: fun(A,nat)] : ~ bij_betw(A,nat,F3,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)
           => ! [F3: fun(A,nat)] : ~ bij_betw(A,nat,F3,S2,top_top(set(nat))) ) ) ) ).

% countable_enum_cases
tff(fact_7673_range__from__nat__into,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( countable_countable(A,A3)
       => ( aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3)),top_top(set(nat))) = A3 ) ) ) ).

% range_from_nat_into
tff(fact_7674_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_7675_from__nat__into__inject,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( countable_countable(A,A3)
       => ( ( B3 != bot_bot(set(A)) )
         => ( countable_countable(A,B3)
           => ( ( aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3) = aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),B3) )
            <=> ( A3 = B3 ) ) ) ) ) ) ).

% from_nat_into_inject
tff(fact_7676_from__nat__into__inj__infinite,axiom,
    ! [A: $tType,A3: set(A),Mb: nat,Nb: nat] :
      ( countable_countable(A,A3)
     => ( ~ aa(set(A),$o,finite_finite2(A),A3)
       => ( ( aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3),Mb) = aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3),Nb) )
        <=> ( Mb = Nb ) ) ) ) ).

% from_nat_into_inj_infinite
tff(fact_7677_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_7678_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_7679_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_7680_to__nat__on__from__nat__into__infinite,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( countable_countable(A,A3)
     => ( ~ aa(set(A),$o,finite_finite2(A),A3)
       => ( aa(A,nat,countable_to_nat_on(A,A3),aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3),Nb)) = Nb ) ) ) ).

% to_nat_on_from_nat_into_infinite
tff(fact_7681_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_7682_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_7683_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_7684_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_7685_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_7686_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_7687_from__nat__into,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( ( A3 != bot_bot(set(A)) )
     => member(A,aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3),Nb),A3) ) ).

% from_nat_into
tff(fact_7688_butlast_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),butlast(A),nil(A)) = nil(A) ).

% butlast.simps(1)
tff(fact_7689_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_7690_map__butlast,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),butlast(B),Xs)) = aa(list(A),list(A),butlast(A),aa(list(B),list(A),map(B,A,F2),Xs)) ).

% map_butlast
tff(fact_7691_inj__on__from__nat__into,axiom,
    ! [A: $tType] : inj_on(set(A),fun(nat,A),counta4804993851260445106t_into(A),collect(set(A),aTP_Lamp_ahh(set(A),$o))) ).

% inj_on_from_nat_into
tff(fact_7692_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_7693_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_7694_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_7695_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_7696_butlast__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xb: A] :
      aa(list(A),list(A),butlast(A),list_update(A,Xs,K,Xb)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),aa(list(A),list(A),butlast(A),Xs),list_update(A,aa(list(A),list(A),butlast(A),Xs),K,Xb)) ).

% butlast_list_update
tff(fact_7697_range__from__nat__into__subset,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( A3 != bot_bot(set(A)) )
     => aa(set(A),$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),A3)),top_top(set(nat)))),A3) ) ).

% range_from_nat_into_subset
tff(fact_7698_subset__range__from__nat__into,axiom,
    ! [A: $tType,A3: set(A)] :
      ( countable_countable(A,A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A3)),top_top(set(nat)))) ) ).

% subset_range_from_nat_into
tff(fact_7699_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_7700_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_7701_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_7702_comp__fun__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
    <=> ( finite4664212375090638736ute_on(A,B,S2,F2)
        & finite4980608107308702382axioms(A,B,S2,F2) ) ) ).

% comp_fun_idem_on_def
tff(fact_7703_comp__fun__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( finite4980608107308702382axioms(A,B,S2,F2)
       => finite673082921795544331dem_on(A,B,S2,F2) ) ) ).

% comp_fun_idem_on.intro
tff(fact_7704_comp__fun__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: 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),F2,X3)),aa(A,fun(B,B),F2,X3)) = aa(A,fun(B,B),F2,X3) ) )
     => finite4980608107308702382axioms(A,B,S2,F2) ) ).

% comp_fun_idem_on_axioms.intro
tff(fact_7705_comp__fun__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite4980608107308702382axioms(A,B,S2,F2)
    <=> ! [X2: A] :
          ( member(A,X2,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X2)),aa(A,fun(B,B),F2,X2)) = aa(A,fun(B,B),F2,X2) ) ) ) ).

% comp_fun_idem_on_axioms_def
tff(fact_7706_comp__fun__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F2)
     => finite4980608107308702382axioms(A,B,S2,F2) ) ).

% comp_fun_idem_on.axioms(2)
tff(fact_7707_shuffles_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,Xb,Xaa) = Y )
     => ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xb),Xaa))
       => ( ( ( Xb = nil(A) )
           => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(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)),product_Pair(list(A),list(A),nil(A)),Xaa)) ) )
         => ( ( ( Xaa = nil(A) )
             => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(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)),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) )
                 => ! [Y3: A,Ys3: list(A)] :
                      ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                     => ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3)),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)),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),Y3),Ys3))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_7708_Field__insert,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2)),R2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))),field2(A,R2)) ).

% Field_insert
tff(fact_7709_Field__empty,axiom,
    ! [A: $tType] : field2(A,bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ).

% Field_empty
tff(fact_7710_shuffles_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),$o))] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),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)),product_Pair(list(A),list(A),nil(A)),Ys3))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),Ys3) )
       => ( ! [Xs2: list(A)] :
              ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs2),nil(A)))
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs2),nil(A)) )
         => ( ! [X3: A,Xs2: list(A),Y3: A,Ys3: list(A)] :
                ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)))
               => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3))
                 => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,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),P,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),Y3),Ys3)) ) ) )
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,A0),A1) ) ) ) ) ).

% shuffles.pinduct
tff(fact_7711_mono__Field,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)),$o),ord_less_eq(set(product_prod(A,A))),R2),S)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),field2(A,R2)),field2(A,S)) ) ).

% mono_Field
tff(fact_7712_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_7713_Field__Restr__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_adl(set(A),fun(A,set(A)),A3))))),A3) ).

% Field_Restr_subset
tff(fact_7714_Field__natLeq__on,axiom,
    ! [Nb: nat] : field2(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Nb)))) = collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Nb)) ).

% Field_natLeq_on
tff(fact_7715_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)),product_Pair(list(A),list(A),Xs),nil(A)))
     => ( shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Xs),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_7716_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)),product_Pair(list(A),list(A),nil(A)),Ys))
     => ( shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),Ys),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_7717_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A)] :
      ( accp(product_prod(list(A),list(A)),shuffles_rel(A),aa(list(A),product_prod(list(A),list(A)),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),Y),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),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys))) ) ) ).

% shuffles.psimps(3)
tff(fact_7718_Refl__Field__Restr2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
      ( refl_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),field2(A,R2))
       => ( field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A3,aTP_Lamp_adl(set(A),fun(A,set(A)),A3)))) = A3 ) ) ) ).

% Refl_Field_Restr2
tff(fact_7719_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_7720_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_7721_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_7722_Order__Relation_OunderS__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A2)),field2(A,R2)) ).

% Order_Relation.underS_Field
tff(fact_7723_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),$o))] :
      ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),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)),product_Pair(list(A),list(A),nil(A)),Ys3))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,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)),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),P,Ys3),Xs2)
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,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),P,A0),A1) ) ) ) ).

% splice.pinduct
tff(fact_7724_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)),insert2(A),A2),bot_bot(set(A)))) ) ) ) ).

% Refl_under_underS
tff(fact_7725_under__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_under(A,R2,A2)),field2(A,R2)) ).

% under_Field
tff(fact_7726_underS__subset__under,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A2)),order_under(A,R2,A2)) ).

% underS_subset_under
tff(fact_7727_splice_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Y: list(A)] :
      ( ( splice(A,Xb,Xaa) = Y )
     => ( accp(product_prod(list(A),list(A)),splice_rel(A),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xb),Xaa))
       => ( ( ( Xb = nil(A) )
           => ( ( Y = Xaa )
             => ~ accp(product_prod(list(A),list(A)),splice_rel(A),aa(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) )
               => ( ( Y = 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)),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_7728_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)),insert2(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))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_7729_splice__Nil2,axiom,
    ! [A: $tType,Xs: list(A)] : splice(A,Xs,nil(A)) = Xs ).

% splice_Nil2
tff(fact_7730_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_7731_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_7732_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_7733_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_7734_splice_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : splice(A,nil(A),Ys) = Ys ).

% splice.simps(1)
tff(fact_7735_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_7736_splice_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Y: list(A)] :
      ( ( splice(A,Xb,Xaa) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != Xaa ) )
       => ~ ! [X3: A,Xs2: list(A)] :
              ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),splice(A,Xaa,Xs2)) ) ) ) ) ).

% splice.elims
tff(fact_7737_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_7738_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),product_Pair(A,A,A2),B2),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_7739_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)),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_7740_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)),product_Pair(list(A),list(A),nil(A)),Ys))
     => ( splice(A,nil(A),Ys) = Ys ) ) ).

% splice.psimps(1)
tff(fact_7741_finite__Linear__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: A,P: fun(A,$o)] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( member(A,Xb,field2(A,R2))
       => ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
         => ( ! [X3: A] :
                ( member(A,X3,field2(A,R2))
               => ( ! [Y2: A] :
                      ( member(A,Y2,order_aboveS(A,R2,X3))
                     => aa(A,$o,P,Y2) )
                 => aa(A,$o,P,X3) ) )
           => aa(A,$o,P,Xb) ) ) ) ) ).

% finite_Linear_order_induct
tff(fact_7742_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))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A4),A4)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_7743_rtrancl__empty,axiom,
    ! [A: $tType] : transitive_rtrancl(A,bot_bot(set(product_prod(A,A)))) = id2(A) ).

% rtrancl_empty
tff(fact_7744_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_7745_bsqr__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R2) = collect(product_prod(product_prod(A,A),product_prod(A,A)),product_case_prod(product_prod(A,A),product_prod(A,A),$o,product_case_prod(A,A,fun(product_prod(A,A),$o),aTP_Lamp_ahj(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),R2)))) ).

% bsqr_def
tff(fact_7746_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B,Y: B,A2: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S2)
       => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),Y)
         => ( member(A,A2,A3)
           => ? [Y7: B] :
                ( ( Y = aa(B,B,aa(A,fun(B,B),F2,A2),Y7) )
                & aa(B,$o,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))),Y7) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_7747_fold__graph_Osimps,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,A1: set(A),A22: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A1),A22)
    <=> ( ( ( A1 = bot_bot(set(A)) )
          & ( A22 = Z ) )
        | ? [X2: A,A8: set(A),Y4: B] :
            ( ( A1 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),A8) )
            & ( A22 = aa(B,B,aa(A,fun(B,B),F2,X2),Y4) )
            & ~ member(A,X2,A8)
            & aa(B,$o,finite_fold_graph(A,B,F2,Z,A8),Y4) ) ) ) ).

% fold_graph.simps
tff(fact_7748_fold__graph_Ocases,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,A1: set(A),A22: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F2,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)),insert2(A),X3),A6) )
             => ! [Y3: B] :
                  ( ( A22 = aa(B,B,aa(A,fun(B,B),F2,X3),Y3) )
                 => ( ~ member(A,X3,A6)
                   => ~ aa(B,$o,finite_fold_graph(A,B,F2,Z,A6),Y3) ) ) ) ) ) ).

% fold_graph.cases
tff(fact_7749_fold__graph_OinsertI,axiom,
    ! [A: $tType,B: $tType,Xb: A,A3: set(A),F2: fun(A,fun(B,B)),Z: B,Y: B] :
      ( ~ member(A,Xb,A3)
     => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),Y)
       => aa(B,$o,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),aa(B,B,aa(A,fun(B,B),F2,Xb),Y)) ) ) ).

% fold_graph.insertI
tff(fact_7750_comp__fun__commute__on_Ofold__graph__finite,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),Z: B,A3: set(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),Y)
       => aa(set(A),$o,finite_finite2(A),A3) ) ) ).

% comp_fun_commute_on.fold_graph_finite
tff(fact_7751_finite__imp__fold__graph,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ? [X_1: B] : aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),X_1) ) ).

% finite_imp_fold_graph
tff(fact_7752_fold__graph__closed__lemma,axiom,
    ! [A: $tType,B: $tType,G: fun(A,fun(B,B)),Z: B,A3: set(A),Xb: B,B3: set(B),F2: fun(A,fun(B,B))] :
      ( aa(B,$o,finite_fold_graph(A,B,G,Z,A3),Xb)
     => ( ! [A4: A,B4: B] :
            ( member(A,A4,A3)
           => ( member(B,B4,B3)
             => ( aa(B,B,aa(A,fun(B,B),F2,A4),B4) = aa(B,B,aa(A,fun(B,B),G,A4),B4) ) ) )
       => ( ! [A4: A,B4: B] :
              ( member(A,A4,A3)
             => ( member(B,B4,B3)
               => member(B,aa(B,B,aa(A,fun(B,B),G,A4),B4),B3) ) )
         => ( member(B,Z,B3)
           => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),Xb)
              & member(B,Xb,B3) ) ) ) ) ) ).

% fold_graph_closed_lemma
tff(fact_7753_fold__graph__closed__eq,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),Z: B] :
      ( ! [A4: A,B4: B] :
          ( member(A,A4,A3)
         => ( member(B,B4,B3)
           => ( aa(B,B,aa(A,fun(B,B),F2,A4),B4) = aa(B,B,aa(A,fun(B,B),G,A4),B4) ) ) )
     => ( ! [A4: A,B4: B] :
            ( member(A,A4,A3)
           => ( member(B,B4,B3)
             => member(B,aa(B,B,aa(A,fun(B,B),G,A4),B4),B3) ) )
       => ( member(B,Z,B3)
         => ( finite_fold_graph(A,B,F2,Z,A3) = finite_fold_graph(A,B,G,Z,A3) ) ) ) ) ).

% fold_graph_closed_eq
tff(fact_7754_empty__fold__graphE,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,Xb: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F2,Z,bot_bot(set(A))),Xb)
     => ( Xb = Z ) ) ).

% empty_fold_graphE
tff(fact_7755_fold__graph_OemptyI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B] : aa(B,$o,finite_fold_graph(A,B,F2,Z,bot_bot(set(A))),Z) ).

% fold_graph.emptyI
tff(fact_7756_comp__fun__commute__on_Ofold__graph__determ,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B,Xb: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S2)
       => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),Xb)
         => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),Y)
           => ( Y = Xb ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_determ
tff(fact_7757_fold__graph__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A3: set(A),F2: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G,A3)
     => ( finite_fold_graph(B,C,F2,Z,aa(set(A),set(B),image2(A,B,G),A3)) = finite_fold_graph(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F2),G),Z,A3) ) ) ).

% fold_graph_image
tff(fact_7758_comp__fun__commute__on_Ofold__graph__insertE,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),Xb: A,A3: set(A),Z: B,V2: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),S2)
       => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),A3)),V2)
         => ( ~ member(A,Xb,A3)
           => ~ ! [Y3: B] :
                  ( ( V2 = aa(B,B,aa(A,fun(B,B),F2,Xb),Y3) )
                 => ~ aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),Y3) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE
tff(fact_7759_comp__fun__commute__on_Ofold__equality,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S2)
       => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),Y)
         => ( finite_fold(A,B,F2,Z,A3) = Y ) ) ) ) ).

% comp_fun_commute_on.fold_equality
tff(fact_7760_Finite__Set_Ofold__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Z: A,A3: set(B)] :
      finite_fold(B,A,F2,Z,A3) = $ite(aa(set(B),$o,finite_finite2(B),A3),the(A,finite_fold_graph(B,A,F2,Z,A3)),Z) ).

% Finite_Set.fold_def
tff(fact_7761_comp__fun__commute__on_Ofold__graph__fold,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,fun(B,B)),A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),S2)
       => ( aa(set(A),$o,finite_finite2(A),A3)
         => aa(B,$o,finite_fold_graph(A,B,F2,Z,A3),finite_fold(A,B,F2,Z,A3)) ) ) ) ).

% comp_fun_commute_on.fold_graph_fold
tff(fact_7762_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)) )
             => ? [X2: A] :
                  ( member(A,X2,A8)
                  & ! [Xa3: A] :
                      ( member(A,Xa3,A8)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X2),Xa3),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_7763_power_Opower__eq__if,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),P3: A,Mb: nat] :
      power2(A,One,Times,P3,Mb) = $ite(Mb = zero_zero(nat),One,aa(A,A,aa(A,fun(A,A),Times,P3),power2(A,One,Times,P3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))) ).

% power.power_eq_if
tff(fact_7764_wf__empty,axiom,
    ! [A: $tType] : wf(A,bot_bot(set(product_prod(A,A)))) ).

% wf_empty
tff(fact_7765_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_7766_wf__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lex(A,R2)) ) ).

% wf_lex
tff(fact_7767_wf__lenlex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lenlex(A,R2)) ) ).

% wf_lenlex
tff(fact_7768_wf__less,axiom,
    wf(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,ord_less(nat)))) ).

% wf_less
tff(fact_7769_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,$o),F2: fun(A,nat),G: fun(A,A)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,aa(A,A,G,X3))),aa(A,nat,F2,X3)) )
     => wf(A,collect(product_prod(A,A),product_case_prod(A,A,$o,aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_ahk(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),P),G)))) ) ).

% wf_if_measure
tff(fact_7770_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,collect(product_prod(A,A),product_case_prod(A,A,$o,ord_less(A)))) ) ).

% wf
tff(fact_7771_wfE__min_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z2: A] :
              ( member(A,Z2,Q)
             => ~ ! [Y2: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y2),Z2),R)
                   => ~ member(A,Y2,Q) ) ) ) ) ).

% wfE_min'
tff(fact_7772_wf__bounded__measure,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F2: fun(A,nat)] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B4),A4),R2)
         => ( aa(nat,$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,$o),ord_less_eq(nat),aa(A,nat,F2,B4)),aa(A,nat,Ub,A4))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,A4)),aa(A,nat,F2,B4)) ) )
     => wf(A,R2) ) ).

% wf_bounded_measure
tff(fact_7773_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_7774_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_7775_wf__int__ge__less__than2,axiom,
    ! [D2: int] : wf(int,int_ge_less_than2(D2)) ).

% wf_int_ge_less_than2
tff(fact_7776_wf__int__ge__less__than,axiom,
    ! [D2: int] : wf(int,int_ge_less_than(D2)) ).

% wf_int_ge_less_than
tff(fact_7777_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_7778_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P: fun(B,$o),K: B,Mb: fun(B,A)] :
      ( wf(A,R2)
     => ( ! [X3: A,Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y3),transitive_trancl(A,R2))
          <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y3),X3),transitive_rtrancl(A,R2)) )
       => ( aa(B,$o,P,K)
         => ? [X3: B] :
              ( aa(B,$o,P,X3)
              & ! [Y2: B] :
                  ( aa(B,$o,P,Y2)
                 => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(B,A,Mb,X3)),aa(B,A,Mb,Y2)),transitive_rtrancl(A,R2)) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_7779_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)) ) )
         => ? [X2: A] :
              ( member(A,X2,A8)
              & ! [Xa3: A] :
                  ( member(A,Xa3,A8)
                 => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X2),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_7780_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,set(B)),F2: fun(A,set(B))] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B4),A4),R2)
         => ( aa(set(B),$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),F2,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),F2,A4)),aa(A,set(B),F2,B4)) ) )
     => wf(A,R2) ) ).

% wf_bounded_set
tff(fact_7781_finite__subset__wf,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => wf(set(A),collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),$o,aTP_Lamp_ahl(set(A),fun(set(A),fun(set(A),$o)),A3)))) ) ).

% finite_subset_wf
tff(fact_7782_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))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A)),R),S2)) ) ) ).

% reduction_pairI
tff(fact_7783_partial__order__on__well__order__on,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A)] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
     => ( order_7125193373082350890der_on(A,A3,R2)
       => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A))) ) ) ).

% partial_order_on_well_order_on
tff(fact_7784_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_7785_finite__Partial__order__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: A,P: fun(A,$o)] :
      ( order_7125193373082350890der_on(A,field2(A,R2),R2)
     => ( member(A,Xb,field2(A,R2))
       => ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
         => ( ! [X3: A] :
                ( member(A,X3,field2(A,R2))
               => ( ! [Y2: A] :
                      ( member(A,Y2,order_aboveS(A,R2,X3))
                     => aa(A,$o,P,Y2) )
                 => aa(A,$o,P,X3) ) )
           => aa(A,$o,P,Xb) ) ) ) ) ).

% finite_Partial_order_induct
tff(fact_7786_rp__inv__image__rp,axiom,
    ! [A: $tType,B: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),F2: fun(B,A)] :
      ( fun_reduction_pair(A,P)
     => 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),P),F2)) ) ).

% rp_inv_image_rp
tff(fact_7787_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)
         => ? [K2: nat] :
            ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),M3)
             => ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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,M3)),aa(nat,rat,X5,N2)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_7788_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_7789_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)
       => ? [K3: nat] :
          ! [M: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),M)
           => ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),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,M)),aa(nat,rat,X5,N4)))),R2) ) ) ) ) ).

% cauchyD
tff(fact_7790_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] :
            ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),M2)
             => ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),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)))),R3) ) ) )
     => cauchy(X5) ) ).

% cauchyI
tff(fact_7791_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)
             => ? [K2: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N2)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X5,N2)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y5,N2)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_7792_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)
            & ? [K3: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),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_7793_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)
          & ! [N: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N))),A9) )
     => ( vanishes(Y5)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ahm(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X5),Y5)) ) ) ).

% vanishes_mult_bounded
tff(fact_7794_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)
         => ? [K2: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N2)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N2))),R5) ) ) ) ).

% vanishes_def
tff(fact_7795_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] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X5,N))),R3) ) )
     => vanishes(X5) ) ).

% vanishesI
tff(fact_7796_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)
       => ? [K3: nat] :
          ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),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_7797_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)
            & ? [K3: nat] :
                ( ! [N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),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),K3),N4)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B4),aa(nat,rat,X5,N4)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_7798_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,$o),ord_less(rat),zero_zero(rat)),R5)
           => ? [K2: nat] :
              ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N2)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X5,N2)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_7799_positive__Real,axiom,
    ! [X5: fun(nat,rat)] :
      ( cauchy(X5)
     => ( aa(real,$o,positive2,real2(X5))
      <=> ? [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
            & ? [K2: nat] :
              ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N2)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,X5,N2)) ) ) ) ) ).

% positive_Real
tff(fact_7800_less__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
    <=> aa(real,$o,positive2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),Xb)) ) ).

% less_real_def
tff(fact_7801_Real_Opositive_Orep__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,positive2,Xb)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K2: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N2)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,aa(real,fun(nat,rat),rep_real,Xb),N2)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_7802_rp__inv__image__def,axiom,
    ! [B: $tType,A: $tType] : fun_rp_inv_image(A,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_ahn(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_7803_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($o)),aTP_Lamp_aho(fun(nat,rat),$o)) ).

% Real.positive_def
tff(fact_7804_finite__def,axiom,
    ! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),$o),aTP_Lamp_yx(fun(set(A),$o),fun(set(A),$o))) ).

% finite_def
tff(fact_7805_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A)] : complete_lattice_lfp(A,F2) = aa(set(A),A,complete_Inf_Inf(A),collect(A,aTP_Lamp_ahp(fun(A,A),fun(A,$o),F2))) ) ).

% lfp_def
tff(fact_7806_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_7807_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,fun(A,A))] :
          ( ! [X3: A,Y3: A,W: A,Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),W)),aa(A,A,aa(A,fun(A,A),F2,Y3),Z2)) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_ahq(fun(A,fun(A,A)),fun(A,A),F2)) = complete_lattice_lfp(A,aTP_Lamp_ahr(fun(A,fun(A,A)),fun(A,A),F2)) ) ) ) ).

% lfp_lfp
tff(fact_7808_lfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),G: fun(A,A)] :
          ( ! [Z8: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,Z8)),aa(A,A,G,Z8))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),complete_lattice_lfp(A,G)) ) ) ).

% lfp_mono
tff(fact_7809_lfp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,A3)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),A3) ) ) ).

% lfp_lowerbound
tff(fact_7810_lfp__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),A3: A] :
          ( ! [U3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,U3)),U3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),U3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),complete_lattice_lfp(A,F2)) ) ) ).

% lfp_greatest
tff(fact_7811_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: A] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F2)),P))),P)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),P) ) ) ) ).

% lfp_induct
tff(fact_7812_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,F2: fun(A,A),P: A] :
          ( ( A3 = complete_lattice_lfp(A,F2) )
         => ( order_mono(A,A,F2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),P))),P)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),P) ) ) ) ) ).

% def_lfp_induct
tff(fact_7813_lfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: fun(A,$o)] :
          ( order_mono(A,A,F2)
         => ( ! [S5: A] :
                ( aa(A,$o,P,S5)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S5),complete_lattice_lfp(A,F2))
                 => aa(A,$o,P,aa(A,A,F2,S5)) ) )
           => ( ! [M7: set(A)] :
                  ( ! [X: A] :
                      ( member(A,X,M7)
                     => aa(A,$o,P,X) )
                 => aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),M7)) )
             => aa(A,$o,P,complete_lattice_lfp(A,F2)) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_7814_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F2)
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2)) = complete_lattice_lfp(A,F2) ) ) ) ).

% lfp_funpow
tff(fact_7815_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( order_mono(A,A,F2)
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F2),bot_bot(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_7816_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: fun(A,A)] : comple7512665784863727008ratesp(A,X) = complete_lattice_lfp(fun(A,$o),aTP_Lamp_zr(fun(A,A),fun(fun(A,$o),fun(A,$o)),X)) ) ).

% iteratesp_def
tff(fact_7817_lfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P: fun(A,$o),F2: fun(A,A),Alpha: fun(A,B),G: fun(B,B)] :
          ( aa(A,$o,P,bot_bot(A))
         => ( ! [X3: A] :
                ( aa(A,$o,P,X3)
               => aa(A,$o,P,aa(A,A,F2,X3)) )
           => ( ! [M7: fun(nat,A)] :
                  ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M7,I3))
                 => aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,M7),top_top(set(nat))))) )
             => ( ! [M7: fun(nat,A)] :
                    ( order_mono(nat,A,M7)
                   => ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M7,I3))
                     => ( 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_ahs(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M7)),top_top(set(nat)))) ) ) )
               => ( order_sup_continuous(A,A,F2)
                 => ( order_sup_continuous(B,B,G)
                   => ( ! [X3: A] :
                          ( aa(A,$o,P,X3)
                         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),complete_lattice_lfp(A,F2))
                           => ( aa(A,B,Alpha,aa(A,A,F2,X3)) = aa(B,B,G,aa(A,B,Alpha,X3)) ) ) )
                     => ( ! [X3: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G,X3))
                       => ( aa(A,B,Alpha,complete_lattice_lfp(A,F2)) = complete_lattice_lfp(B,G) ) ) ) ) ) ) ) ) ) ) ).

% lfp_transfer_bounded
tff(fact_7818_lfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Alpha: fun(A,B),F2: fun(A,A),G: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( order_sup_continuous(A,A,F2)
           => ( order_sup_continuous(B,B,G)
             => ( ! [X3: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G,X3))
               => ( ! [X3: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),complete_lattice_lfp(A,F2))
                     => ( aa(A,B,Alpha,aa(A,A,F2,X3)) = aa(B,B,G,aa(A,B,Alpha,X3)) ) )
                 => ( aa(A,B,Alpha,complete_lattice_lfp(A,F2)) = complete_lattice_lfp(B,G) ) ) ) ) ) ) ) ).

% lfp_transfer
tff(fact_7819_cclfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta3822494911875563373attice(A) )
     => ! [Alpha: fun(A,B),F2: fun(A,A),G: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( order_mono(A,A,F2)
           => ( ( aa(A,B,Alpha,bot_bot(A)) = bot_bot(B) )
             => ( ! [X3: A] : aa(A,B,Alpha,aa(A,A,F2,X3)) = aa(B,B,G,aa(A,B,Alpha,X3))
               => ( aa(A,B,Alpha,order_532582986084564980_cclfp(A,F2)) = order_532582986084564980_cclfp(B,G) ) ) ) ) ) ) ).

% cclfp_transfer
tff(fact_7820_iteratesp_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),A2: A] :
          ( aa(A,$o,comple7512665784863727008ratesp(A,F2),A2)
        <=> ( ? [X2: A] :
                ( ( A2 = aa(A,A,F2,X2) )
                & aa(A,$o,comple7512665784863727008ratesp(A,F2),X2) )
            | ? [M8: set(A)] :
                ( ( A2 = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X2: A] :
                    ( member(A,X2,M8)
                   => aa(A,$o,comple7512665784863727008ratesp(A,F2),X2) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_7821_iteratesp_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),A2: A] :
          ( aa(A,$o,comple7512665784863727008ratesp(A,F2),A2)
         => ( ! [X3: A] :
                ( ( A2 = aa(A,A,F2,X3) )
               => ~ aa(A,$o,comple7512665784863727008ratesp(A,F2),X3) )
           => ~ ! [M7: set(A)] :
                  ( ( A2 = aa(set(A),A,complete_Sup_Sup(A),M7) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M7)
                   => ~ ! [X: A] :
                          ( member(A,X,M7)
                         => aa(A,$o,comple7512665784863727008ratesp(A,F2),X) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_7822_iteratesp_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M4: set(A),F2: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M4)
         => ( ! [X3: A] :
                ( member(A,X3,M4)
               => aa(A,$o,comple7512665784863727008ratesp(A,F2),X3) )
           => aa(A,$o,comple7512665784863727008ratesp(A,F2),aa(set(A),A,complete_Sup_Sup(A),M4)) ) ) ) ).

% iteratesp.Sup
tff(fact_7823_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_aht(fun(A,A),fun(nat,A),F4)),top_top(set(nat)))) ) ) ) ).

% sup_continuous_lfp
tff(fact_7824_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_acg(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ) ).

% ord_class.lexordp_def
tff(fact_7825_finite__refines__card__le,axiom,
    ! [A: $tType,A3: set(A),R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A3,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,A3,R)
         => ( equiv_equiv(A,A3,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,A3,S2))),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A3,R))) ) ) ) ) ).

% finite_refines_card_le
tff(fact_7826_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_7827_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_7828_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A),Y: A,Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
              & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_7829_in__quotient__imp__subset,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( equiv_equiv(A,A3,R2)
     => ( member(set(A),X5,equiv_quotient(A,A3,R2))
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),A3) ) ) ).

% in_quotient_imp_subset
tff(fact_7830_in__quotient__imp__non__empty,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A)] :
      ( equiv_equiv(A,A3,R2)
     => ( member(set(A),X5,equiv_quotient(A,A3,R2))
       => ( X5 != bot_bot(set(A)) ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_7831_equiv__listrel,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] :
      ( equiv_equiv(A,A3,R2)
     => equiv_equiv(list(A),lists(A,A3),listrel(A,A,R2)) ) ).

% equiv_listrel
tff(fact_7832_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_7833_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_7834_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_7835_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Xs: list(A),Ys: list(A)] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
           => ( aa(list(A),$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),Y),Ys)) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_7836_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Xs: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(list(A),$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),Y),Ys)) ) ) ).

% lexordp.Cons
tff(fact_7837_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_7838_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_7839_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_7840_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_7841_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_7842_lexordp_ONil,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Y: A,Ys: list(A)] : aa(list(A),$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),Y),Ys)) ) ).

% lexordp.Nil
tff(fact_7843_quotient__disj,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A),Y5: set(A)] :
      ( equiv_equiv(A,A3,R2)
     => ( member(set(A),X5,equiv_quotient(A,A3,R2))
       => ( member(set(A),Y5,equiv_quotient(A,A3,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_7844_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
         => ( ( ( A1 = nil(A) )
             => ! [Y3: A,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
           => ( ! [X3: A] :
                  ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)
                 => ! [Y3: A] :
                      ( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3) ) )
             => ~ ! [X3: A,Y3: A,Xs2: list(A)] :
                    ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                   => ! [Ys3: list(A)] :
                        ( ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
                       => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3)
                         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X3)
                           => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_7845_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A1: list(A),A22: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A1),A22)
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( A1 = nil(A) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) ) )
            | ? [X2: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4) )
            | ? [X2: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
                & ( A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X2)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs3),Ys4) ) ) ) ) ).

% lexordp.simps
tff(fact_7846_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
         => ( ( ( Xs = nil(A) )
             => ! [Y3: A,Ys5: list(A)] : Ys != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
           => ( ! [X3: A] :
                  ( ? [Xs4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4)
                 => ! [Y3: A] :
                      ( ? [Ys5: list(A)] : Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3) ) )
             => ~ ! [X3: A,Xs4: list(A)] :
                    ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
                   => ! [Ys5: list(A)] :
                        ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys5) )
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs4),Ys5) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_7847_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A),P: fun(list(A),fun(list(A),$o))] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
         => ( ! [Y3: A,Ys3: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3))
           => ( ! [X3: A,Xs2: list(A),Y3: A,Ys3: list(A)] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y3)
                 => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,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),Y3),Ys3)) )
             => ( ! [X3: A,Xs2: list(A),Ys3: list(A)] :
                    ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3)
                   => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs2),Ys3)
                     => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),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),P,Xs),Ys) ) ) ) ) ) ).

% lexordp_induct
tff(fact_7848_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
        <=> ( ? [X2: A,Vs3: list(A)] : Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Vs3))
            | ? [Us2: list(A),A7: A,B7: A,Vs3: list(A),Ws3: list(A)] :
                ( aa(A,$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),Vs3)) )
                & ( 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_7849_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A,Us: list(A),Xs: list(A),Ys: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(list(A),$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),Y),Ys))) ) ) ).

% lexordp_append_left_rightI
tff(fact_7850_finite__refines__finite,axiom,
    ! [A: $tType,A3: set(A),R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A3,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,A3,R)
         => ( equiv_equiv(A,A3,S2)
           => aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A3,S2)) ) ) ) ) ).

% finite_refines_finite
tff(fact_7851_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),Xb: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( member(A,Xb,A3)
       => ( member(A,Y,A3)
         => ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))),R2) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))),R2) )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_7852_equiv__imp__dvd__card,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( equiv_equiv(A,A3,R2)
       => ( ! [X6: set(A)] :
              ( member(set(A),X6,equiv_quotient(A,A3,R2))
             => dvd_dvd(nat,K,aa(set(A),nat,finite_card(A),X6)) )
         => dvd_dvd(nat,K,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% equiv_imp_dvd_card
tff(fact_7853_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),X5: set(A),Xb: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( member(set(A),X5,equiv_quotient(A,A3,R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),X5)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_7854_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys)
        <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys),lexord(A,collect(product_prod(A,A),product_case_prod(A,A,$o,ord_less(A))))) ) ) ).

% lexordp_conv_lexord
tff(fact_7855_proj__iff,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),Xb: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),A3)
       => ( ( equiv_proj(A,A,R2,Xb) = equiv_proj(A,A,R2,Y) )
        <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2) ) ) ) ).

% proj_iff
tff(fact_7856_ord_Olexordp__def,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : lexordp2(A,Less) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_acf(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Less)) ).

% ord.lexordp_def
tff(fact_7857_ord_Olexordp__simps_I3_J,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xb: A,Xs: list(A),Y: 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),Y),Ys))
    <=> ( aa(A,$o,aa(A,fun(A,$o),Less,Xb),Y)
        | ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y),Xb)
          & aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs),Ys) ) ) ) ).

% ord.lexordp_simps(3)
tff(fact_7858_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_7859_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_7860_ord_Olexordp_ONil,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Y: 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),Y),Ys)) ).

% ord.lexordp.Nil
tff(fact_7861_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) )
         => ! [Y3: A,Ys3: list(A)] : A22 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
       => ( ! [X3: A] :
              ( ? [Xs2: list(A)] : A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)
             => ! [Y3: A] :
                  ( ? [Ys3: list(A)] : A22 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3)
                 => ~ aa(A,$o,aa(A,fun(A,$o),Less,X3),Y3) ) )
         => ~ ! [X3: A,Y3: 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),Y3),Ys3) )
                   => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,X3),Y3)
                     => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y3),X3)
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs2),Ys3) ) ) ) ) ) ) ) ).

% ord.lexordp.cases
tff(fact_7862_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) ) )
        | ? [X2: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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,X2),Y4) )
        | ? [X2: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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,X2),Y4)
            & ~ aa(A,$o,aa(A,fun(A,$o),Less,Y4),X2)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),lexordp2(A,Less),Xs3),Ys4) ) ) ) ).

% ord.lexordp.simps
tff(fact_7863_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_7864_ord_Olexordp_Ocong,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : lexordp2(A,Less) = lexordp2(A,Less) ).

% ord.lexordp.cong
tff(fact_7865_ord_Olexordp_OCons__eq,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xb: A,Y: A,Xs: list(A),Ys: list(A)] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Xb),Y)
     => ( ~ aa(A,$o,aa(A,fun(A,$o),Less,Y),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),Y),Ys)) ) ) ) ).

% ord.lexordp.Cons_eq
tff(fact_7866_ord_Olexordp_OCons,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xb: A,Y: A,Xs: list(A),Ys: list(A)] :
      ( aa(A,$o,aa(A,fun(A,$o),Less,Xb),Y)
     => 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),Y),Ys)) ) ).

% ord.lexordp.Cons
tff(fact_7867_ord_Olexordp__append__left__rightI,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o)),Xb: A,Y: A,Us: list(A),Xs: list(A),Ys: list(A)] :
      ( aa(A,$o,aa(A,fun(A,$o),Less,Xb),Y)
     => 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),Y),Ys))) ) ).

% ord.lexordp_append_left_rightI
tff(fact_7868_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_7869_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_7870_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_7871_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_7872_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,Xb: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xb,Xaa,Xba,Xc) = Y )
     => ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),Xb),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xaa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xba),Xc))))
       => ~ ( ( Y = $ite(aa(nat,$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))) )
           => ~ accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),Xb),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xaa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xba),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7873_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_7874_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_7875_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_7876_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_7877_enumerate__Ex,axiom,
    ! [S2: set(nat),S: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => ( member(nat,S,S2)
       => ? [N: nat] : aa(nat,nat,infini527867602293511546merate(nat,S2),N) = S ) ) ).

% enumerate_Ex
tff(fact_7878_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_7879_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_7880_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_7881_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_7882_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)
           => ? [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(set(A),nat,finite_card(A),S2))
                & ( aa(nat,A,infini527867602293511546merate(A,S2),N) = S ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_7883_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X5: set(A),Y5: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X5))
             => ( aa(nat,A,infini527867602293511546merate(A,X5),I2) = aa(nat,A,infini527867602293511546merate(A,Y5),I2) ) )
         => ( aa(set(A),$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_7884_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_7885_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_7886_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_7887_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_7888_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_7889_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A1),aa(A,product_prod(nat,A),product_Pair(nat,A,A22),A32))))
     => ( ! [F3: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc: A] :
            ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A4),aa(A,product_prod(nat,A),product_Pair(nat,A,B4),Acc))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B4),A4)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F3,A4),Acc)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F3),A4),B4),Acc) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7890_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_7891_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_7892_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)),insert2(A),aa(nat,A,infini527867602293511546merate(A,S2),zero_zero(nat))),bot_bot(set(A))))),Nb) ) ).

% enumerate_Suc'
tff(fact_7893_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X5: set(A),Y5: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X5))
             => ( aa(nat,A,infini527867602293511546merate(A,X5),I2) = aa(nat,A,infini527867602293511546merate(A,Y5),I2) ) )
         => ( aa(set(A),$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_7894_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      ( accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A2),aa(A,product_prod(nat,A),product_Pair(nat,A,B2),Acc2))))
     => ( set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F2,A2),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_7895_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,$o),aTP_Lamp_ahu(set(A),fun(nat,fun(A,$o)),S2),Nb)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_7896_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] : aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Nb)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),ord_Least(A,aTP_Lamp_ahv(set(A),fun(A,$o),S2))),bot_bot(set(A))))),Nb) ) ).

% enumerate_Suc
tff(fact_7897_Least__eq__0,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_7898_Least__Suc,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ahw(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_7899_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% LeastI
tff(fact_7900_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
         => ( ! [X3: A] :
                ( aa(A,$o,P,X3)
               => aa(A,$o,Q,X3) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_ex
tff(fact_7901_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% LeastI_ex
tff(fact_7902_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A2)
         => ( ! [X3: A] :
                ( aa(A,$o,P,X3)
               => aa(A,$o,Q,X3) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2
tff(fact_7903_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),ord_Least(A,P))
         => ~ aa(A,$o,P,K) ) ) ).

% not_less_Least
tff(fact_7904_Least__Suc2,axiom,
    ! [P: fun(nat,$o),Nb: nat,Q: fun(nat,$o),Mb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,Q,Mb)
       => ( ~ aa(nat,$o,P,zero_zero(nat))
         => ( ! [K3: nat] :
                ( aa(nat,$o,P,aa(nat,nat,suc,K3))
              <=> aa(nat,$o,Q,K3) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7905_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
         => ( ! [A4: A] :
                ( aa(A,$o,P,A4)
               => ( ! [B10: A] :
                      ( aa(A,$o,P,B10)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B10) )
                 => aa(A,$o,Q,A4) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_7906_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A2)
         => ( ! [A4: A] :
                ( aa(A,$o,P,A4)
               => ( ! [B10: A] :
                      ( aa(A,$o,P,B10)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B10) )
                 => aa(A,$o,Q,A4) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder
tff(fact_7907_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y3) )
           => ( ord_Least(A,P) = Xb ) ) ) ) ).

% Least_equality
tff(fact_7908_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y3) )
           => ( ! [X3: A] :
                  ( aa(A,$o,P,X3)
                 => ( ! [Y2: A] :
                        ( aa(A,$o,P,Y2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2) )
                   => aa(A,$o,Q,X3) ) )
             => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ) ).

% LeastI2_order
tff(fact_7909_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Z: A] :
          ( ? [X: A] :
              ( aa(A,$o,P,X)
              & ! [Y3: A] :
                  ( aa(A,$o,P,Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
              & ! [Y3: A] :
                  ( ( aa(A,$o,P,Y3)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Ya2) ) )
                 => ( Y3 = X ) ) )
         => ( aa(A,$o,P,Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),Z) ) ) ) ).

% Least1_le
tff(fact_7910_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] :
          ( ? [X: A] :
              ( aa(A,$o,P,X)
              & ! [Y3: A] :
                  ( aa(A,$o,P,Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
              & ! [Y3: A] :
                  ( ( aa(A,$o,P,Y3)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Ya2) ) )
                 => ( Y3 = X ) ) )
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% Least1I
tff(fact_7911_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),K) ) ) ).

% Least_le
tff(fact_7912_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,P))
         => ( ? [X_12: A] : aa(A,$o,P,X_12)
           => ( ord_Least(A,P) = aa(set(A),A,lattic643756798350308766er_Min(A),collect(A,P)) ) ) ) ) ).

% Least_Min
tff(fact_7913_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S2: set(A),P: fun(A,$o)] : abort_Bleast(A,S2,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahx(set(A),fun(fun(A,$o),fun(A,$o)),S2),P)) ) ).

% abort_Bleast_def
tff(fact_7914_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S2: set(A),P: fun(A,$o)] : bleast(A,S2,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahx(set(A),fun(fun(A,$o),fun(A,$o)),S2),P)) ) ).

% Bleast_def
tff(fact_7915_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] : aa(nat,A,infini527867602293511546merate(A,S2),zero_zero(nat)) = ord_Least(A,aTP_Lamp_ahv(set(A),fun(A,$o),S2)) ) ).

% enumerate_0
tff(fact_7916_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),S2: set(A)] :
          ( order_mono(A,B,F2)
         => ( ? [X: A] :
                ( member(A,X,S2)
                & ! [Xa2: A] :
                    ( member(A,Xa2,S2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2) ) )
           => ( ord_Least(B,aa(set(A),fun(B,$o),aTP_Lamp_ahy(fun(A,B),fun(set(A),fun(B,$o)),F2),S2)) = aa(A,B,F2,ord_Least(A,aTP_Lamp_ahz(set(A),fun(A,$o),S2))) ) ) ) ) ).

% Least_mono
tff(fact_7917_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_aia(fun(A,$o),fun(A,$o),P)) ) ).

% Least_def
tff(fact_7918_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Nb: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Nb)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_ahu(set(A),fun(nat,fun(A,$o)),S2),Nb)) ) ) ) ).

% enumerate_Suc''
tff(fact_7919_stable__sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Sk: fun(fun(A,B),fun(list(A),list(A)))] :
          ( linord3483353639454293061rt_key(A,B,Sk)
        <=> ! [F7: fun(A,B),Xs3: list(A),K2: B] : aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,$o)),F7),K2)),aa(list(A),list(A),aa(fun(A,B),fun(list(A),list(A)),Sk,F7),Xs3)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,$o),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,$o)),F7),K2)),Xs3) ) ) ).

% stable_sort_key_def
tff(fact_7920_independentD,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),Ta: set(A),U: fun(A,real),V2: A] :
          ( ~ real_V358717886546972837endent(A,S)
         => ( aa(set(A),$o,finite_finite2(A),Ta)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S)
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),U)),Ta) = zero_zero(A) )
               => ( member(A,V2,Ta)
                 => ( aa(A,real,U,V2) = zero_zero(real) ) ) ) ) ) ) ) ).

% independentD
tff(fact_7921_independent__empty,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ~ real_V358717886546972837endent(A,bot_bot(set(A))) ) ).

% independent_empty
tff(fact_7922_dependent__single,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] :
          ( real_V358717886546972837endent(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))
        <=> ( Xb = zero_zero(A) ) ) ) ).

% dependent_single
tff(fact_7923_dependent__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),A3: set(A)] :
          ( real_V358717886546972837endent(A,B3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
           => real_V358717886546972837endent(A,A3) ) ) ) ).

% dependent_mono
tff(fact_7924_independent__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ~ real_V358717886546972837endent(A,A3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
           => ~ real_V358717886546972837endent(A,B3) ) ) ) ).

% independent_mono
tff(fact_7925_independent__Union__directed,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [C5: set(set(A))] :
          ( ! [C4: set(A),D4: set(A)] :
              ( member(set(A),C4,C5)
             => ( member(set(A),D4,C5)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),D4)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D4),C4) ) ) )
         => ( ! [C4: set(A)] :
                ( member(set(A),C4,C5)
               => ~ real_V358717886546972837endent(A,C4) )
           => ~ real_V358717886546972837endent(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)) ) ) ) ).

% independent_Union_directed
tff(fact_7926_dependent__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( member(A,zero_zero(A),A3)
         => real_V358717886546972837endent(A,A3) ) ) ).

% dependent_zero
tff(fact_7927_unique__representation,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( ! [V3: A] :
                ( ( aa(A,real,F2,V3) != zero_zero(real) )
               => member(A,V3,Basis) )
           => ( ! [V3: A] :
                  ( ( aa(A,real,G,V3) != zero_zero(real) )
                 => member(A,V3,Basis) )
             => ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),F2)))
               => ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),G)))
                 => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),F2)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),F2))) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),G)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),G))) )
                   => ( F2 = G ) ) ) ) ) ) ) ) ).

% unique_representation
tff(fact_7928_independent__if__scalars__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [F3: fun(A,real),X3: A] :
                ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),F3)),A3) = zero_zero(A) )
               => ( member(A,X3,A3)
                 => ( aa(A,real,F3,X3) = zero_zero(real) ) ) )
           => ~ real_V358717886546972837endent(A,A3) ) ) ) ).

% independent_if_scalars_zero
tff(fact_7929_dependent__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( real_V358717886546972837endent(A,S2)
          <=> ? [U6: fun(A,real)] :
                ( ? [X2: A] :
                    ( member(A,X2,S2)
                    & ( aa(A,real,U6,X2) != zero_zero(real) ) )
                & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),U6)),S2) = zero_zero(A) ) ) ) ) ) ).

% dependent_finite
tff(fact_7930_independentD__unique,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),X5: fun(A,real),Y5: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,B3)
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X5)))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X5))),B3)
             => ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),Y5)))
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),Y5))),B3)
                 => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),X5)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X5))) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),Y5)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),Y5))) )
                   => ( X5 = Y5 ) ) ) ) ) ) ) ) ).

% independentD_unique
tff(fact_7931_stable__sort__key__sort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => linord3483353639454293061rt_key(A,B,linorder_sort_key(A,B)) ) ).

% stable_sort_key_sort_key
tff(fact_7932_independent__explicit__finite__subsets,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: set(A)] :
          ( ~ real_V358717886546972837endent(A,A3)
        <=> ! [S9: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S9),A3)
             => ( aa(set(A),$o,finite_finite2(A),S9)
               => ! [U6: fun(A,real)] :
                    ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),U6)),S9) = zero_zero(A) )
                   => ! [X2: A] :
                        ( member(A,X2,S9)
                       => ( aa(A,real,U6,X2) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_finite_subsets
tff(fact_7933_independent__explicit__module,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( ~ real_V358717886546972837endent(A,S)
        <=> ! [T2: set(A),U6: fun(A,real),V7: A] :
              ( aa(set(A),$o,finite_finite2(A),T2)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),S)
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),U6)),T2) = zero_zero(A) )
                 => ( member(A,V7,T2)
                   => ( aa(A,real,U6,V7) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_module
tff(fact_7934_dependent__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( real_V358717886546972837endent(A,S)
        <=> ? [T2: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),T2)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),S)
              & ? [U6: fun(A,real)] :
                  ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),U6)),T2) = zero_zero(A) )
                  & ? [X2: A] :
                      ( member(A,X2,T2)
                      & ( aa(A,real,U6,X2) != zero_zero(real) ) ) ) ) ) ) ).

% dependent_explicit
tff(fact_7935_independentD__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A),X5: fun(A,real),Xb: A] :
          ( ~ real_V358717886546972837endent(A,B3)
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X5)))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X5))),B3)
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),X5)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X5))) = zero_zero(A) )
               => ( aa(A,real,X5,Xb) = zero_zero(real) ) ) ) ) ) ) ).

% independentD_alt
tff(fact_7936_independent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] :
          ( ~ real_V358717886546972837endent(A,B3)
        <=> ! [X8: fun(A,real)] :
              ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X8)))
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X8))),B3)
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),X8)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X8))) = zero_zero(A) )
                 => ! [X2: A] : aa(A,real,X8,X2) = zero_zero(real) ) ) ) ) ) ).

% independent_alt
tff(fact_7937_dependent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] :
          ( real_V358717886546972837endent(A,B3)
        <=> ? [X8: fun(A,real)] :
              ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X8)))
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X8))),B3)
              & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aib(fun(A,real),fun(A,A),X8)),collect(A,aTP_Lamp_aic(fun(A,real),fun(A,$o),X8))) = zero_zero(A) )
              & ? [X2: A] : aa(A,real,X8,X2) != zero_zero(real) ) ) ) ).

% dependent_alt
tff(fact_7938_isUCont__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,top_top(set(A)),F2)
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X2: A,Y4: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Y4)),S6)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X2),aa(A,B,F2,Y4))),R5) ) ) ) ) ) ).

% isUCont_def
tff(fact_7939_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Nb: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,Nb)
        <=> ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_7940_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat)) ) ).

% possible_bit_0
tff(fact_7941_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Ia: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,Ia)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),Ia)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_7942_uniformly__continuous__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [S: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( member(A,X2,S)
                     => ! [Xa3: A] :
                          ( member(A,Xa3,S)
                         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa3,X2)),D6)
                           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,Xa3),aa(A,B,F2,X2))),E3) ) ) ) ) ) ) ) ).

% uniformly_continuous_on_def
tff(fact_7943_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Mb),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
                & bit_se6407376104438227557le_bit(A,type2(A),Nb) ))),
            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))) ) ).

% drop_bit_exp_eq
tff(fact_7944_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% bit_minus_2_iff
tff(fact_7945_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_7946_of__nat__CHAR,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),semiri4206861660011772517g_char(A,type2(A))) = zero_zero(A) ) ) ).

% of_nat_CHAR
tff(fact_7947_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,Mb)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% bit_mask_iff
tff(fact_7948_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X3: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X3) = zero_zero(A) )
               => dvd_dvd(nat,C2,X3) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_7949_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Nb) = zero_zero(A) )
        <=> dvd_dvd(nat,semiri4206861660011772517g_char(A,type2(A)),Nb) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_7950_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
            & ( aa(nat,A,semiring_1_of_nat(A),N2) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7951_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X3)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),C2)
                   => ( aa(nat,A,semiring_1_of_nat(A),X3) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7952_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( aa(nat,A,semiring_1_of_nat(A),N2) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7953_bit__push__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_se4730199178511100633sh_bit(A,Mb),A2)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
            & bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ) ).

% bit_push_bit_iff
tff(fact_7954_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% fold_possible_bit
tff(fact_7955_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Mb))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7956_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(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))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7957_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))
            & ( Nb != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ) ).

% bit_double_iff
tff(fact_7958_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),product_case_prod(A,B,fun(set(B),set(B)),aTP_Lamp_aid(set(A),fun(A,fun(B,fun(set(B),set(B)))),S2)),bot_bot(set(B)),R) ) ) ).

% Image_fold
tff(fact_7959_map__rec,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F2),Xs) = aa(list(B),list(A),rec_list(list(A),B,nil(A),aTP_Lamp_aie(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F2)),Xs) ).

% map_rec
tff(fact_7960_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_7961_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_7962_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)),insert2(B),A2),bot_bot(set(B)))))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,A2),B2),R2) ) ).

% Image_singleton_iff
tff(fact_7963_listrel__Nil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,A))] : image(list(B),list(A),listrel(B,A,R2),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),nil(B)),bot_bot(set(list(B))))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listrel_Nil
tff(fact_7964_countable__Image,axiom,
    ! [B: $tType,A: $tType,Y5: set(A),X5: set(product_prod(A,B))] :
      ( ! [Y3: A] :
          ( member(A,Y3,Y5)
         => countable_countable(B,image(A,B,X5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A))))) )
     => ( countable_countable(A,Y5)
       => countable_countable(B,image(A,B,X5,Y5)) ) ) ).

% countable_Image
tff(fact_7965_Image__INT__subset,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),B3: fun(C,set(B)),A3: set(C)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,R2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B3),A3)))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_aif(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B3)),A3))) ).

% Image_INT_subset
tff(fact_7966_Image__subset,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),A3: set(A),B3: set(B),C5: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,R2,C5)),B3) ) ).

% Image_subset
tff(fact_7967_Image__Int__subset,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A3: set(B),B3: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(B,A,R,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(B,A,R,A3)),image(B,A,R,B3))) ).

% Image_Int_subset
tff(fact_7968_finite__rtrancl__Image,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: set(A)] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(set(A),$o,finite_finite2(A),A3)
       => aa(set(A),$o,finite_finite2(A),image(A,A,transitive_rtrancl(A,R),A3)) ) ) ).

% finite_rtrancl_Image
tff(fact_7969_list_Osimps_I6_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,fun(list(B),fun(A,A)))] : aa(list(B),A,rec_list(A,B,F1,F22),nil(B)) = F1 ).

% list.simps(6)
tff(fact_7970_finite__Image,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),A3: set(A)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => aa(set(B),$o,finite_finite2(B),image(A,B,R,A3)) ) ).

% finite_Image
tff(fact_7971_list_Osimps_I7_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,fun(list(B),fun(A,A))),X21: B,X222: list(B)] : aa(list(B),A,rec_list(A,B,F1,F22),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X21),X222)) = aa(A,A,aa(list(B),fun(A,A),aa(B,fun(list(B),fun(A,A)),F22,X21),X222),aa(list(B),A,rec_list(A,B,F1,F22),X222)) ).

% list.simps(7)
tff(fact_7972_Image__closed__trancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,X5)),X5)
     => ( image(A,A,transitive_rtrancl(A,R2),X5) = X5 ) ) ).

% Image_closed_trancl
tff(fact_7973_Image__mono,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,B)),R2: set(product_prod(A,B)),A11: set(A),A3: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R4),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),A3)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,R4,A11)),image(A,B,R2,A3)) ) ) ).

% Image_mono
tff(fact_7974_quotientE,axiom,
    ! [A: $tType,X5: set(A),A3: set(A),R2: set(product_prod(A,A))] :
      ( member(set(A),X5,equiv_quotient(A,A3,R2))
     => ~ ! [X3: A] :
            ( ( X5 = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A)))) )
           => ~ member(A,X3,A3) ) ) ).

% quotientE
tff(fact_7975_quotientI,axiom,
    ! [A: $tType,Xb: A,A3: set(A),R2: set(product_prod(A,A))] :
      ( member(A,Xb,A3)
     => member(set(A),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))),equiv_quotient(A,A3,R2)) ) ).

% quotientI
tff(fact_7976_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A2: B] : image(B,A,R2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B)))) = collect(A,aa(B,fun(A,$o),aTP_Lamp_aig(set(product_prod(B,A)),fun(B,fun(A,$o)),R2),A2)) ).

% Image_singleton
tff(fact_7977_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_7978_wfE__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: set(A)] :
      ( wf(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),image(A,A,R,A3))
       => ( A3 = bot_bot(set(A)) ) ) ) ).

% wfE_pf
tff(fact_7979_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)),insert2(B),Xb),bot_bot(set(B)))) ).

% proj_def
tff(fact_7980_equiv__class__self,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( member(A,A2,A3)
       => member(A,A2,image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) ) ) ).

% equiv_class_self
tff(fact_7981_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))),insert2(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)),insert2(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))),insert2(list(B)),Xs),bot_bot(set(list(B)))))) ).

% listrel_Cons
tff(fact_7982_equiv__class__eq__iff,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),Xb: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2)
      <=> ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
          & member(A,Xb,A3)
          & member(A,Y,A3) ) ) ) ).

% equiv_class_eq_iff
tff(fact_7983_eq__equiv__class__iff,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),Xb: A,Y: A] :
      ( equiv_equiv(A,A3,R2)
     => ( member(A,Xb,A3)
       => ( member(A,Y,A3)
         => ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))) )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xb),Y),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_7984_equiv__class__eq,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),R2)
       => ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_7985_eq__equiv__class,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A,A3: set(A)] :
      ( ( image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A3,R2)
       => ( member(A,B2,A3)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),R2) ) ) ) ).

% eq_equiv_class
tff(fact_7986_refines__equiv__class__eq2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A)),A3: set(A),A2: A] :
      ( aa(set(product_prod(A,A)),$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,A3,R)
       => ( equiv_equiv(A,A3,S2)
         => ( image(A,A,S2,image(A,A,R,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_7987_refines__equiv__class__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A)),A3: set(A),A2: A] :
      ( aa(set(product_prod(A,A)),$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,A3,R)
       => ( equiv_equiv(A,A3,S2)
         => ( image(A,A,R,image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A))))) = image(A,A,S2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_7988_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)),insert2(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_7989_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),B3: set(B)] : image(B,A,R2,B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aih(set(product_prod(B,A)),fun(B,set(A)),R2)),B3)) ).

% Image_eq_UN
tff(fact_7990_subset__equiv__class,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),B2: A,A2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( aa(set(A),$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)),insert2(A),B2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))
       => ( member(A,B2,A3)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),R2) ) ) ) ).

% subset_equiv_class
tff(fact_7991_equiv__class__subset,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),R2)
       => aa(set(A),$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)),insert2(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))) ) ) ).

% equiv_class_subset
tff(fact_7992_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),Xb: A,A2: A,B2: A] :
      ( equiv_equiv(A,A3,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)),insert2(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A))))))
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_7993_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)),insert2(A),Xb),bot_bot(set(A))),R2) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% singleton_quotient
tff(fact_7994_list_Orec__o__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: B,Ga: fun(C,fun(list(C),fun(B,B))),F2: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),B),comp(list(C),B,list(A),rec_list(B,C,G,Ga)),map(A,C,F2)) = rec_list(B,A,G,aa(fun(A,C),fun(A,fun(list(A),fun(B,B))),aTP_Lamp_aii(fun(C,fun(list(C),fun(B,B))),fun(fun(A,C),fun(A,fun(list(A),fun(B,B)))),Ga),F2)) ).

% list.rec_o_map
tff(fact_7995_quotient__def,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A))] : equiv_quotient(A,A3,R2) = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(A),set(set(set(A))),image2(A,set(set(A)),aTP_Lamp_aij(set(product_prod(A,A)),fun(A,set(set(A))),R2)),A3)) ).

% quotient_def
tff(fact_7996_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)),F2: 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,F2)
         => ( 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_aik(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F2),A22)),image(A,A,R1,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F2,A1),A22) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_7997_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,A3: set(A),R2: set(product_prod(A,A)),F2: fun(A,set(B)),A2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( equiv_congruent(A,set(B),R2,F2)
       => ( member(A,A2,A3)
         => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))))) = aa(A,set(B),F2,A2) ) ) ) ) ).

% UN_equiv_class
tff(fact_7998_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)),F2: 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,F2)
         => ( 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_aik(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F2),A2)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_7999_set__rec,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),rec_list(set(A),A,bot_bot(set(A)),aTP_Lamp_ail(A,fun(list(A),fun(set(A),set(A))))),Xs) ).

% set_rec
tff(fact_8000_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)),insert2(A),A2),bot_bot(set(A))))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B2),A2),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_8001_subset__Image__Image__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: set(A),B3: set(A)] :
      ( order_preorder_on(A,field2(A,R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),field2(A,R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),field2(A,R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),image(A,A,R2,A3)),image(A,A,R2,B3))
          <=> ! [X2: A] :
                ( member(A,X2,A3)
               => ? [Xa3: A] :
                    ( member(A,Xa3,B3)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X2),R2) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_8002_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_8003_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)),insert2(A),A2),bot_bot(set(A)))) = image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))) )
            <=> ( A2 = B2 ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_8004_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
            $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_8005_antisym__empty,axiom,
    ! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).

% antisym_empty
tff(fact_8006_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_8007_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_8008_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xaa),Xb) ) ).

% less_integer.abs_eq
tff(fact_8009_sgn__integer__code,axiom,
    ! [K: code_integer] :
      sgn_sgn(code_integer,K) = $ite(
        K = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_8010_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))),insert2(product_prod(A,A)),Xb),bot_bot(set(product_prod(A,A))))) ).

% antisym_singleton
tff(fact_8011_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_8012_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,product_case_prod(code_integer,code_integer,int,aTP_Lamp_aim(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ) ).

% int_of_integer_code
tff(fact_8013_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_ain(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% nat_of_integer_code
tff(fact_8014_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer))
     => ( code_nat_of_integer(K) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_8015_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_8016_less__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Xb),Xaa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_integer.rep_eq
tff(fact_8017_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_zero(code_integer)) = zero_zero(nat) ).

% nat_of_integer_code_post(1)
tff(fact_8018_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),product_Pair(code_integer,$o,zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o),aTP_Lamp_aio(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% bit_cut_integer_code
tff(fact_8019_append__butlast__last__id,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),butlast(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,Xs)),nil(A))) = Xs ) ) ).

% append_butlast_last_id
tff(fact_8020_last__remdups__adj,axiom,
    ! [A: $tType,Xs: list(A)] : last(A,remdups_adj(A,Xs)) = last(A,Xs) ).

% last_remdups_adj
tff(fact_8021_last__appendL,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( ( Ys = nil(A) )
     => ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Xs) ) ) ).

% last_appendL
tff(fact_8022_last__appendR,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( ( Ys != nil(A) )
     => ( last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = last(A,Ys) ) ) ).

% last_appendR
tff(fact_8023_last__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( last(A,replicate(A,Nb,Xb)) = Xb ) ) ).

% last_replicate
tff(fact_8024_last__upt,axiom,
    ! [Ia: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ia),J)
     => ( last(nat,upt(Ia,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_8025_last__snoc,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)))) = Xb ).

% last_snoc
tff(fact_8026_last__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))
     => ( last(A,drop(A,Nb,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_8027_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
         => ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),product_Pair(A,B,last(A,Xs)),last(B,Ys)) ) ) ) ) ).

% last_zip
tff(fact_8028_last__rev,axiom,
    ! [A: $tType,Xs: list(A)] : last(A,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),A,hd(A),Xs) ).

% last_rev
tff(fact_8029_hd__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),rev(A),Xs)) = last(A,Xs) ).

% hd_rev
tff(fact_8030_last__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( ( Xs = nil(A) )
        | ( aa(list(A),list(A),tl(A),Xs) != nil(A) ) )
     => ( last(A,aa(list(A),list(A),tl(A),Xs)) = last(A,Xs) ) ) ).

% last_tl
tff(fact_8031_hd__Nil__eq__last,axiom,
    ! [A: $tType] : aa(list(A),A,hd(A),nil(A)) = last(A,nil(A)) ).

% hd_Nil_eq_last
tff(fact_8032_last__in__set,axiom,
    ! [A: $tType,As: list(A)] :
      ( ( As != nil(A) )
     => member(A,last(A,As),aa(list(A),set(A),set2(A),As)) ) ).

% last_in_set
tff(fact_8033_longest__common__suffix,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
    ? [Ss: list(A),Xs4: list(A),Ys5: list(A)] :
      ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),Ss) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),Ss) )
      & ( ( Xs4 = nil(A) )
        | ( Ys5 = nil(A) )
        | ( last(A,Xs4) != last(A,Ys5) ) ) ) ).

% longest_common_suffix
tff(fact_8034_last__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      last(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(Ys = nil(A),last(A,Xs),last(A,Ys)) ).

% last_append
tff(fact_8035_last_Osimps,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(Xs = nil(A),Xb,last(A,Xs)) ).

% last.simps
tff(fact_8036_last__ConsL,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ( Xs = nil(A) )
     => ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = Xb ) ) ).

% last_ConsL
tff(fact_8037_last__ConsR,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ( Xs != nil(A) )
     => ( last(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = last(A,Xs) ) ) ).

% last_ConsR
tff(fact_8038_last__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F2: fun(A,B)] :
      ( ( Xs != nil(A) )
     => ( last(B,aa(list(A),list(B),map(A,B,F2),Xs)) = aa(A,B,F2,last(A,Xs)) ) ) ).

% last_map
tff(fact_8039_dropWhile__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => ( last(A,dropWhile(A,P,Xs)) = last(A,Xs) ) ) ) ).

% dropWhile_last
tff(fact_8040_takeWhile__not__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( takeWhile(A,aTP_Lamp_aip(list(A),fun(A,$o),Xs),Xs) = aa(list(A),list(A),butlast(A),Xs) ) ) ).

% takeWhile_not_last
tff(fact_8041_snoc__eq__iff__butlast,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) = Ys )
    <=> ( ( Ys != nil(A) )
        & ( aa(list(A),list(A),butlast(A),Ys) = Xs )
        & ( last(A,Ys) = Xb ) ) ) ).

% snoc_eq_iff_butlast
tff(fact_8042_remdups__adj__append_H,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( ( Xs = nil(A) )
        | ( Ys = nil(A) )
        | ( last(A,Xs) != aa(list(A),A,hd(A),Ys) ) )
     => ( remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,Xs)),remdups_adj(A,Ys)) ) ) ).

% remdups_adj_append'
tff(fact_8043_remdups__adj__append_H_H,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,Xs)),remdups_adj(A,dropWhile(A,aTP_Lamp_aiq(list(A),fun(A,$o),Xs),Ys))) ) ) ).

% remdups_adj_append''
tff(fact_8044_last__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( last(A,Xs) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_8045_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,Xb: A] :
      ( ( Xs != nil(A) )
     => ( last(A,list_update(A,Xs,K,Xb)) = $ite(K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xb,last(A,Xs)) ) ) ).

% last_list_update
tff(fact_8046_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_air(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),K),
            product_apsnd(code_integer,code_integer,code_integer,uminus_uminus(code_integer),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_ais(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_8047_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un2480387367778600638or_num(Mb,Nb)) ) ).

% numeral_xor_num
tff(fact_8048_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num] :
          ( ( bit_un2480387367778600638or_num(Mb,Nb) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_8049_ran__map__add,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,M1,M22)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,M1)),ran(A,B,M22)) ) ) ).

% ran_map_add
tff(fact_8050_List_Ounion__def,axiom,
    ! [A: $tType] : union(A) = fold(A,list(A),insert(A)) ).

% List.union_def
tff(fact_8051_in__set__insert,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),Xs) = Xs ) ) ).

% in_set_insert
tff(fact_8052_distinct__insert,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_insert
tff(fact_8053_insert__Nil,axiom,
    ! [A: $tType,Xb: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) ).

% insert_Nil
tff(fact_8054_not__in__set__insert,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) ) ) ).

% not_in_set_insert
tff(fact_8055_List_Oset__insert,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),aa(list(A),set(A),set2(A),Xs)) ).

% List.set_insert
tff(fact_8056_insert__remdups,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),remdups(A,Xs)) = remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),Xs)) ).

% insert_remdups
tff(fact_8057_map__add__comm,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( map_add(A,B,M1,M22) = map_add(A,B,M22,M1) ) ) ).

% map_add_comm
tff(fact_8058_List_Oinsert__def,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),Xs) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) ).

% List.insert_def
tff(fact_8059_remove__code_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xb),coset(A,Xs)) = coset(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),insert(A),Xb),Xs)) ).

% remove_code(2)
tff(fact_8060_finite__range__map__of__map__add,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),L: list(product_prod(B,A))] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),F2),top_top(set(B))))
     => aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(B),set(option(A)),image2(B,option(A),map_add(B,A,F2,map_of(B,A,L))),top_top(set(B)))) ) ).

% finite_range_map_of_map_add
tff(fact_8061_graph__map__add,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( graph(A,B,map_add(A,B,M1,M22)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),graph(A,B,M1)),graph(A,B,M22)) ) ) ).

% graph_map_add
tff(fact_8062_inv__image__partition,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) )
     => ( ! [Y3: A] :
            ( member(A,Y3,aa(list(A),set(A),set2(A),Ys))
           => ~ aa(A,$o,P,Y3) )
       => ( vimage(list(A),product_prod(list(A),list(A)),partition(A,P),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),insert2(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),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_8063_vimage__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B3: set(B)] :
      ( member(A,A2,vimage(A,B,F2,B3))
    <=> member(B,aa(A,B,F2,A2),B3) ) ).

% vimage_eq
tff(fact_8064_vimageI,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: B,B2: A,B3: set(A)] :
      ( ( aa(B,A,F2,A2) = B2 )
     => ( member(A,B2,B3)
       => member(B,A2,vimage(B,A,F2,B3)) ) ) ).

% vimageI
tff(fact_8065_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),P: fun(B,$o)] : vimage(A,B,F2,collect(B,P)) = collect(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_ait(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P)) ).

% vimage_Collect_eq
tff(fact_8066_vimage__ident,axiom,
    ! [A: $tType,Y5: set(A)] : vimage(A,A,aTP_Lamp_lg(A,A),Y5) = Y5 ).

% vimage_ident
tff(fact_8067_vimage__UNIV,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : vimage(A,B,F2,top_top(set(B))) = top_top(set(A)) ).

% vimage_UNIV
tff(fact_8068_vimage__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : vimage(A,B,F2,bot_bot(set(B))) = bot_bot(set(A)) ).

% vimage_empty
tff(fact_8069_vimage__Int,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B3)) ).

% vimage_Int
tff(fact_8070_vimage__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B3)) ).

% vimage_Un
tff(fact_8071_vimage__const,axiom,
    ! [B: $tType,A: $tType,C2: B,A3: set(B)] :
      vimage(A,B,aTP_Lamp_lv(B,fun(A,B),C2),A3) = $ite(member(B,C2,A3),top_top(set(A)),bot_bot(set(A))) ).

% vimage_const
tff(fact_8072_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_aca(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_8073_image__vimage__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(A)] : aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ).

% image_vimage_eq
tff(fact_8074_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_8075_vimage__if,axiom,
    ! [B: $tType,A: $tType,B3: set(A),C2: B,D2: B,A3: set(B)] :
      vimage(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_aiu(set(A),fun(B,fun(B,fun(A,B))),B3),C2),D2),A3) = $ite(
        member(B,C2,A3),
        $ite(member(B,D2,A3),top_top(set(A)),B3),
        $ite(member(B,D2,A3),aa(set(A),set(A),uminus_uminus(set(A)),B3),bot_bot(set(A))) ) ).

% vimage_if
tff(fact_8076_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B),B3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),vimage(A,B,F2,A3)),vimage(A,B,F2,B3)) ).

% vimage_Diff
tff(fact_8077_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),uminus_uminus(set(B)),A3)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F2,A3)) ).

% vimage_Compl
tff(fact_8078_surj__vimage__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => ( ( vimage(B,A,F2,A3) = bot_bot(set(B)) )
      <=> ( A3 = bot_bot(set(A)) ) ) ) ).

% surj_vimage_empty
tff(fact_8079_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_8080_vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A)] : vimage(A,B,F2,aa(set(A),set(B),image2(A,B,F2),A3)) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_aiv(fun(A,B),fun(set(A),fun(A,$o)),F2),A3)) ).

% vimage_image_eq
tff(fact_8081_vimage__inter__cong,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F2: fun(A,B),G: fun(A,B),Y: set(B)] :
      ( ! [W: A] :
          ( member(A,W,S2)
         => ( aa(A,B,F2,W) = aa(A,B,G,W) ) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,Y)),S2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,G,Y)),S2) ) ) ).

% vimage_inter_cong
tff(fact_8082_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_8083_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A2: B,B3: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),B3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))),vimage(A,B,F2,B3)) ).

% vimage_insert
tff(fact_8084_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B2: B] :
      ( member(A,A2,vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B2),bot_bot(set(B)))))
    <=> ( aa(A,B,F2,A2) = B2 ) ) ).

% vimage_singleton_eq
tff(fact_8085_vimage__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B3: set(B)] : vimage(A,B,F2,B3) = collect(A,aa(set(B),fun(A,$o),aTP_Lamp_aiw(fun(A,B),fun(set(B),fun(A,$o)),F2),B3)) ).

% vimage_def
tff(fact_8086_vimage__Collect,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),F2: fun(A,B),Q: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(B,$o,P,aa(A,B,F2,X3))
        <=> aa(A,$o,Q,X3) )
     => ( vimage(A,B,F2,collect(B,P)) = collect(A,Q) ) ) ).

% vimage_Collect
tff(fact_8087_vimageI2,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: B,A3: set(A)] :
      ( member(A,aa(B,A,F2,A2),A3)
     => member(B,A2,vimage(B,A,F2,A3)) ) ).

% vimageI2
tff(fact_8088_vimageE,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),B3: set(B)] :
      ( member(A,A2,vimage(A,B,F2,B3))
     => member(B,aa(A,B,F2,A2),B3) ) ).

% vimageE
tff(fact_8089_vimageD,axiom,
    ! [A: $tType,B: $tType,A2: A,F2: fun(A,B),A3: set(B)] :
      ( member(A,A2,vimage(A,B,F2,A3))
     => member(B,aa(A,B,F2,A2),A3) ) ).

% vimageD
tff(fact_8090_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_8091_finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( inj_on(B,A,H,A3)
       => 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)),A3)) ) ) ).

% finite_vimage_IntI
tff(fact_8092_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_8093_inf__img__fin__dom,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
     => ( ~ aa(set(B),$o,finite_finite2(B),A3)
       => ? [X3: A] :
            ( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A3))
            & ~ aa(set(B),$o,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_dom
tff(fact_8094_inf__img__fin__domE,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
     => ( ~ aa(set(B),$o,finite_finite2(B),A3)
       => ~ ! [Y3: A] :
              ( member(A,Y3,aa(set(B),set(A),image2(B,A,F2),A3))
             => aa(set(B),$o,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_domE
tff(fact_8095_vimage__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),vimage(B,A,F2,A3)),vimage(B,A,F2,B3)) ) ).

% vimage_mono
tff(fact_8096_subset__vimage__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),vimage(A,B,F2,B3))
    <=> ! [X2: A] :
          ( member(A,X2,A3)
         => member(B,aa(A,B,F2,X2),B3) ) ) ).

% subset_vimage_iff
tff(fact_8097_continuous__imp__open__vimage,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),F2: fun(A,B),B3: set(B)] :
          ( topolo81223032696312382ous_on(A,B,S,F2)
         => ( topolo1002775350975398744n_open(A,S)
           => ( topolo1002775350975398744n_open(B,B3)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,B3)),S)
               => topolo1002775350975398744n_open(A,vimage(A,B,F2,B3)) ) ) ) ) ) ).

% continuous_imp_open_vimage
tff(fact_8098_finite__vimageD_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),vimage(A,B,F2,A3))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),aa(set(A),set(B),image2(A,B,F2),top_top(set(A))))
       => aa(set(B),$o,finite_finite2(B),A3) ) ) ).

% finite_vimageD'
tff(fact_8099_vimage__subsetD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),B3: set(A),A3: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),vimage(B,A,F2,B3)),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),A3)) ) ) ).

% vimage_subsetD
tff(fact_8100_image__vimage__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,A3))),A3) ).

% image_vimage_subset
tff(fact_8101_image__subset__iff__subset__vimage,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A3)),B3)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),vimage(B,A,F2,B3)) ) ).

% image_subset_iff_subset_vimage
tff(fact_8102_vimage__subsetI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(B),A3: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F2),A3))
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,B3)),A3) ) ) ).

% vimage_subsetI
tff(fact_8103_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,Xb: B,A3: set(B),F2: fun(B,set(A))] :
      vimage(A,product_prod(B,A),product_Pair(B,A,Xb),product_Sigma(B,A,A3,F2)) = $ite(member(B,Xb,A3),aa(B,set(A),F2,Xb),bot_bot(set(A))) ).

% Pair_vimage_Sigma
tff(fact_8104_finite__finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F4: set(A),H: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ! [Y3: A] :
            ( member(A,Y3,F4)
           => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A))))),A3)) )
       => aa(set(B),$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)),A3)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_8105_countable__vimage,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
     => ( countable_countable(B,vimage(B,A,F2,B3))
       => countable_countable(A,B3) ) ) ).

% countable_vimage
tff(fact_8106_vimage__subset__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B3: set(B),A3: set(A)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,B3)),A3)
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B3),aa(set(A),set(B),image2(A,B,F2),A3)) ) ) ).

% vimage_subset_eq
tff(fact_8107_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B3: set(B)] : vimage(A,B,F2,B3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_aix(fun(A,B),fun(B,set(A)),F2)),B3)) ).

% vimage_eq_UN
tff(fact_8108_inf__img__fin__domE_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
     => ( ~ aa(set(B),$o,finite_finite2(B),A3)
       => ~ ! [Y3: A] :
              ( member(A,Y3,aa(set(B),set(A),image2(B,A,F2),A3))
             => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y3),bot_bot(set(A))))),A3)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_8109_inf__img__fin__dom_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A3))
     => ( ~ aa(set(B),$o,finite_finite2(B),A3)
       => ? [X3: A] :
            ( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A3))
            & ~ aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))),A3)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_8110_card__vimage__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),aa(set(A),set(B),image2(A,B,F2),top_top(set(A))))
       => ( aa(set(A),nat,finite_card(A),vimage(A,B,F2,A3)) = aa(set(B),nat,finite_card(B),A3) ) ) ) ).

% card_vimage_inj
tff(fact_8111_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),D5: set(A),A3: set(B)] :
      ( inj_on(A,B,F2,D5)
     => ( aa(set(B),$o,finite_finite2(B),A3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,A3)),D5))),aa(set(B),nat,finite_card(B),A3)) ) ) ).

% card_vimage_inj_on_le
tff(fact_8112_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),A3: set(C)] : aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_aiy(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G)),A3) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F2),A3),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_aiz(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F2),G),A3)) ).

% image_split_eq_Sigma
tff(fact_8113_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: B] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,$o),aTP_Lamp_aja(fun(A,B),fun(B,fun(A,$o)),F2),A2))),bot_bot(set(A)))) ) ).

% inj_vimage_singleton
tff(fact_8114_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),A2: B] :
      ( inj_on(A,B,F2,A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A2),bot_bot(set(B))))),A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_ajb(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F2),A3),A2))),bot_bot(set(A)))) ) ).

% inj_on_vimage_singleton
tff(fact_8115_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un7362597486090784418nd_num(Mb,Nb)) ) ).

% numeral_and_num
tff(fact_8116_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ( semiring_gcd_Gcd_fin(A,A3) = zero_zero(A) )
        <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))))
            & aa(set(A),$o,finite_finite2(A),A3) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_8117_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_8118_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A3)
         => ( semiring_gcd_Gcd_fin(A,A3) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_8119_is__unit__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A)] :
          ( dvd_dvd(A,semiring_gcd_Gcd_fin(A,A3),one_one(A))
        <=> ( semiring_gcd_Gcd_fin(A,A3) = one_one(A) ) ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_8120_Gcd__fin__eq__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( semiring_gcd_Gcd_fin(A,A3) = gcd_Gcd(A,A3) ) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_8121_vimage__Suc__insert__0,axiom,
    ! [A3: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),A3)) = vimage(nat,nat,suc,A3) ).

% vimage_Suc_insert_0
tff(fact_8122_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_8123_dvd__gcd__list__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,Xs: list(A)] :
          ( dvd_dvd(A,B2,semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Xs)))
        <=> ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
             => dvd_dvd(A,B2,X2) ) ) ) ).

% dvd_gcd_list_iff
tff(fact_8124_gcd__list__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A2: A] :
          ( ! [B4: A] :
              ( member(A,B4,aa(list(A),set(A),set2(A),Bs))
             => dvd_dvd(A,A2,B4) )
         => dvd_dvd(A,A2,semiring_gcd_Gcd_fin(A,aa(list(A),set(A),set2(A),Bs))) ) ) ).

% gcd_list_greatest
tff(fact_8125_Gcd__fin__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( member(A,A2,A3)
         => dvd_dvd(A,semiring_gcd_Gcd_fin(A,A3),A2) ) ) ).

% Gcd_fin_dvd
tff(fact_8126_dvd__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),B2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( dvd_dvd(A,B2,semiring_gcd_Gcd_fin(A,A3))
          <=> ! [X2: A] :
                ( member(A,X2,A3)
               => dvd_dvd(A,B2,X2) ) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_8127_Gcd__fin__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A3)
         => ( ! [B4: A] :
                ( member(A,B4,A3)
               => dvd_dvd(A,A2,B4) )
           => dvd_dvd(A,A2,semiring_gcd_Gcd_fin(A,A3)) ) ) ) ).

% Gcd_fin_greatest
tff(fact_8128_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num] :
          ( ( bit_un7362597486090784418nd_num(Mb,Nb) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_8129_connected__closed,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo1966860045006549960nected(A,S)
        <=> ~ ? [A8: set(A),B11: set(A)] :
                ( topolo7761053866217962861closed(A,A8)
                & topolo7761053866217962861closed(A,B11)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A8),B11))
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),B11)),S) = bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),S) != bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B11),S) != bot_bot(set(A)) ) ) ) ) ).

% connected_closed
tff(fact_8130_connected__closedD,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),A3: set(A),B3: set(A)] :
          ( topolo1966860045006549960nected(A,S)
         => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),S) = bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
             => ( topolo7761053866217962861closed(A,A3)
               => ( topolo7761053866217962861closed(A,B3)
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),S) = bot_bot(set(A)) )
                    | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),S) = bot_bot(set(A)) ) ) ) ) ) ) ) ) ).

% connected_closedD
tff(fact_8131_connected__Times__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S2: set(A),T3: set(B)] :
          ( topolo1966860045006549960nected(product_prod(A,B),product_Sigma(A,B,S2,aTP_Lamp_ajc(set(B),fun(A,set(B)),T3)))
        <=> ( ( S2 = bot_bot(set(A)) )
            | ( T3 = bot_bot(set(B)) )
            | ( topolo1966860045006549960nected(A,S2)
              & topolo1966860045006549960nected(B,T3) ) ) ) ) ).

% connected_Times_eq
tff(fact_8132_connected__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1966860045006549960nected(A,bot_bot(set(A))) ) ).

% connected_empty
tff(fact_8133_connected__contains__Ioo,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: set(A),A2: A,B2: A] :
          ( topolo1966860045006549960nected(A,A3)
         => ( member(A,A2,A3)
           => ( member(A,B2,A3)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),A3) ) ) ) ) ).

% connected_contains_Ioo
tff(fact_8134_connectedD__interval,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [U2: set(A),Xb: A,Y: A,Z: A] :
          ( topolo1966860045006549960nected(A,U2)
         => ( member(A,Xb,U2)
           => ( member(A,Y,U2)
             => ( aa(A,$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),Y)
                 => member(A,Z,U2) ) ) ) ) ) ) ).

% connectedD_interval
tff(fact_8135_connectedI__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U2: set(A)] :
          ( ! [X3: A,Y3: A,Z2: A] :
              ( member(A,X3,U2)
             => ( member(A,Y3,U2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y3)
                   => member(A,Z2,U2) ) ) ) )
         => topolo1966860045006549960nected(A,U2) ) ) ).

% connectedI_interval
tff(fact_8136_connected__iff__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U2: set(A)] :
          ( topolo1966860045006549960nected(A,U2)
        <=> ! [X2: A] :
              ( member(A,X2,U2)
             => ! [Xa3: A] :
                  ( member(A,Xa3,U2)
                 => ! [Z6: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z6)
                     => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z6),Xa3)
                       => member(A,Z6,U2) ) ) ) ) ) ) ).

% connected_iff_interval
tff(fact_8137_connected__contains__Icc,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: set(A),A2: A,B2: A] :
          ( topolo1966860045006549960nected(A,A3)
         => ( member(A,A2,A3)
           => ( member(A,B2,A3)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),A3) ) ) ) ) ).

% connected_contains_Icc
tff(fact_8138_connected__sing,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A] : topolo1966860045006549960nected(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),bot_bot(set(A)))) ) ).

% connected_sing
tff(fact_8139_connected__Un,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),Ta: set(A)] :
          ( topolo1966860045006549960nected(A,S)
         => ( topolo1966860045006549960nected(A,Ta)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),Ta) != bot_bot(set(A)) )
             => topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),Ta)) ) ) ) ) ).

% connected_Un
tff(fact_8140_connected__Union,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(set(A))] :
          ( ! [S3: set(A)] :
              ( member(set(A),S3,S2)
             => topolo1966860045006549960nected(A,S3) )
         => ( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S2) != bot_bot(set(A)) )
           => topolo1966860045006549960nected(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S2)) ) ) ) ).

% connected_Union
tff(fact_8141_not__in__connected__cases,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),Xb: A] :
          ( topolo1966860045006549960nected(A,S2)
         => ( ~ member(A,Xb,S2)
           => ( ( S2 != bot_bot(set(A)) )
             => ( ( condit941137186595557371_above(A,S2)
                 => ~ ! [Y2: A] :
                        ( member(A,Y2,S2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Xb) ) )
               => ~ ( condit1013018076250108175_below(A,S2)
                   => ~ ! [Y2: A] :
                          ( member(A,Y2,S2)
                         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y2) ) ) ) ) ) ) ) ).

% not_in_connected_cases
tff(fact_8142_connected__diff__open__from__closed,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),Ta: set(A),U: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),Ta)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),U)
           => ( topolo1002775350975398744n_open(A,S)
             => ( topolo7761053866217962861closed(A,Ta)
               => ( topolo1966860045006549960nected(A,U)
                 => ( topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Ta),S))
                   => topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),U),S)) ) ) ) ) ) ) ) ).

% connected_diff_open_from_closed
tff(fact_8143_connectedD,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: set(A),U2: set(A),V: set(A)] :
          ( topolo1966860045006549960nected(A,A3)
         => ( topolo1002775350975398744n_open(A,U2)
           => ( topolo1002775350975398744n_open(A,V)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),V)),A3) = bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),U2),V))
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),A3) = bot_bot(set(A)) )
                    | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),V),A3) = bot_bot(set(A)) ) ) ) ) ) ) ) ) ).

% connectedD
tff(fact_8144_connectedI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U2: set(A)] :
          ( ! [A6: set(A)] :
              ( topolo1002775350975398744n_open(A,A6)
             => ! [B9: set(A)] :
                  ( topolo1002775350975398744n_open(A,B9)
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),U2) != bot_bot(set(A)) )
                   => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B9),U2) != bot_bot(set(A)) )
                     => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B9)),U2) = bot_bot(set(A)) )
                       => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),U2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B9)) ) ) ) ) )
         => topolo1966860045006549960nected(A,U2) ) ) ).

% connectedI
tff(fact_8145_connected__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo1966860045006549960nected(A,S2)
        <=> ~ ? [A8: set(A),B11: set(A)] :
                ( topolo1002775350975398744n_open(A,A8)
                & topolo1002775350975398744n_open(A,B11)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A8),B11))
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),B11)),S2) = bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),S2) != bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B11),S2) != bot_bot(set(A)) ) ) ) ) ).

% connected_def
tff(fact_8146_rat__less__code,axiom,
    ! [P3: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P3),Q2)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_aje(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_8147_disjnt__equiv__class,axiom,
    ! [A: $tType,A3: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( equiv_equiv(A,A3,R2)
     => ( disjnt(A,image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),bot_bot(set(A)))),image(A,A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B2),bot_bot(set(A)))))
      <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2),R2) ) ) ).

% disjnt_equiv_class
tff(fact_8148_disjnt__self__iff__empty,axiom,
    ! [A: $tType,S2: set(A)] :
      ( disjnt(A,S2,S2)
    <=> ( S2 = bot_bot(set(A)) ) ) ).

% disjnt_self_iff_empty
tff(fact_8149_disjnt__insert1,axiom,
    ! [A: $tType,A2: A,X5: set(A),Y5: set(A)] :
      ( disjnt(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X5),Y5)
    <=> ( ~ member(A,A2,Y5)
        & disjnt(A,X5,Y5) ) ) ).

% disjnt_insert1
tff(fact_8150_disjnt__insert2,axiom,
    ! [A: $tType,Y5: set(A),A2: A,X5: set(A)] :
      ( disjnt(A,Y5,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A2),X5))
    <=> ( ~ member(A,A2,Y5)
        & disjnt(A,Y5,X5) ) ) ).

% disjnt_insert2
tff(fact_8151_disjnt__Un1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( disjnt(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3),C5)
    <=> ( disjnt(A,A3,C5)
        & disjnt(A,B3,C5) ) ) ).

% disjnt_Un1
tff(fact_8152_disjnt__Un2,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B3: set(A)] :
      ( disjnt(A,C5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))
    <=> ( disjnt(A,C5,A3)
        & disjnt(A,C5,B3) ) ) ).

% disjnt_Un2
tff(fact_8153_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C5: set(A),A3: set(B),B3: set(B)] :
      ( disjnt(product_prod(A,B),product_Sigma(A,B,C5,aTP_Lamp_adi(set(B),fun(A,set(B)),A3)),product_Sigma(A,B,C5,aTP_Lamp_adi(set(B),fun(A,set(B)),B3)))
    <=> ( ( C5 = bot_bot(set(A)) )
        | disjnt(B,A3,B3) ) ) ).

% disjnt_Times1_iff
tff(fact_8154_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C5: set(B),B3: set(A)] :
      ( disjnt(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_adi(set(B),fun(A,set(B)),C5)),product_Sigma(A,B,B3,aTP_Lamp_adi(set(B),fun(A,set(B)),C5)))
    <=> ( ( C5 = bot_bot(set(B)) )
        | disjnt(A,A3,B3) ) ) ).

% disjnt_Times2_iff
tff(fact_8155_disjnt__insert,axiom,
    ! [A: $tType,Xb: A,N3: set(A),M4: set(A)] :
      ( ~ member(A,Xb,N3)
     => ( disjnt(A,M4,N3)
       => disjnt(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),M4),N3) ) ) ).

% disjnt_insert
tff(fact_8156_disjnt__subset2,axiom,
    ! [A: $tType,X5: set(A),Y5: set(A),Z4: set(A)] :
      ( disjnt(A,X5,Y5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z4),Y5)
       => disjnt(A,X5,Z4) ) ) ).

% disjnt_subset2
tff(fact_8157_disjnt__subset1,axiom,
    ! [A: $tType,X5: set(A),Y5: set(A),Z4: set(A)] :
      ( disjnt(A,X5,Y5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z4),X5)
       => disjnt(A,Z4,Y5) ) ) ).

% disjnt_subset1
tff(fact_8158_disjnt__empty1,axiom,
    ! [A: $tType,A3: set(A)] : disjnt(A,bot_bot(set(A)),A3) ).

% disjnt_empty1
tff(fact_8159_disjnt__empty2,axiom,
    ! [A: $tType,A3: set(A)] : disjnt(A,A3,bot_bot(set(A))) ).

% disjnt_empty2
tff(fact_8160_disjnt__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( disjnt(A,A3,B3)
    <=> ! [X2: A] :
          ~ ( member(A,X2,A3)
            & member(A,X2,B3) ) ) ).

% disjnt_iff
tff(fact_8161_disjnt__sym,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( disjnt(A,A3,B3)
     => disjnt(A,B3,A3) ) ).

% disjnt_sym
tff(fact_8162_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C5: fun(A,set(B)),B3: set(A)] :
      ( disjnt(product_prod(A,B),product_Sigma(A,B,A3,C5),product_Sigma(A,B,B3,C5))
    <=> ( ! [X2: A] :
            ( member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))
           => ( aa(A,set(B),C5,X2) = bot_bot(set(B)) ) )
        | disjnt(A,A3,B3) ) ) ).

% disjnt_Sigma_iff
tff(fact_8163_disjnt__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( disjnt(A,A3,B3)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) ) ) ).

% disjnt_def
tff(fact_8164_disjnt__ge__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y5: set(A),X5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Y5)
         => ( ! [X3: A] :
                ( member(A,X3,X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),Y5)),X3) )
           => disjnt(A,X5,Y5) ) ) ) ).

% disjnt_ge_max
tff(fact_8165_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q2: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q2) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q2) ) ).

% quotient_of_denom_pos
tff(fact_8166_card__Un__disjnt,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( aa(set(A),$o,finite_finite2(A),B3)
       => ( disjnt(A,A3,B3)
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)) ) ) ) ) ).

% card_Un_disjnt
tff(fact_8167_sum__card__image,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( pairwise(A,aTP_Lamp_ajf(fun(A,set(B)),fun(A,fun(A,$o)),F2),A3)
       => ( aa(set(set(B)),nat,aa(fun(set(B),nat),fun(set(set(B)),nat),groups7311177749621191930dd_sum(set(B),nat),finite_card(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),A3)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_na(fun(A,set(B)),fun(A,nat),F2)),A3) ) ) ) ).

% sum_card_image
tff(fact_8168_infinite__infinite__partition,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A3)
     => ~ ! [C8: fun(nat,set(A))] :
            ( pairwise(nat,aTP_Lamp_ajg(fun(nat,set(A)),fun(nat,fun(nat,$o)),C8),top_top(set(nat)))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),C8),top_top(set(nat))))),A3)
             => ~ ! [I3: nat] : ~ aa(set(A),$o,finite_finite2(A),aa(nat,set(A),C8,I3)) ) ) ) ).

% infinite_infinite_partition
tff(fact_8169_pairwise__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B),P: fun(B,fun(B,$o))] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,A3)
         => ( member(A,Y3,A3)
           => ( ( X3 != Y3 )
             => ( ( aa(A,B,F2,X3) != aa(A,B,F2,Y3) )
               => aa(B,$o,aa(B,fun(B,$o),P,aa(A,B,F2,X3)),aa(A,B,F2,Y3)) ) ) ) )
     => pairwise(B,P,aa(set(A),set(B),image2(A,B,F2),A3)) ) ).

% pairwise_imageI
tff(fact_8170_pairwise__image,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,$o)),F2: fun(B,A),S: set(B)] :
      ( pairwise(A,R2,aa(set(B),set(A),image2(B,A,F2),S))
    <=> pairwise(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_ajh(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R2),F2),S) ) ).

% pairwise_image
tff(fact_8171_pairwise__trivial,axiom,
    ! [A: $tType,I5: set(A)] : pairwise(A,aTP_Lamp_afm(A,fun(A,$o)),I5) ).

% pairwise_trivial
tff(fact_8172_pairwise__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S2: set(A)] :
      ( pairwise(A,R,S2)
    <=> ! [X2: A] :
          ( member(A,X2,S2)
         => ! [Xa3: A] :
              ( member(A,Xa3,S2)
             => ( ( X2 != Xa3 )
               => aa(A,$o,aa(A,fun(A,$o),R,X2),Xa3) ) ) ) ) ).

% pairwise_def
tff(fact_8173_pairwiseI,axiom,
    ! [A: $tType,S2: set(A),R: fun(A,fun(A,$o))] :
      ( ! [X3: A,Y3: A] :
          ( member(A,X3,S2)
         => ( member(A,Y3,S2)
           => ( ( X3 != Y3 )
             => aa(A,$o,aa(A,fun(A,$o),R,X3),Y3) ) ) )
     => pairwise(A,R,S2) ) ).

% pairwiseI
tff(fact_8174_pairwiseD,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S2: set(A),Xb: A,Y: A] :
      ( pairwise(A,R,S2)
     => ( member(A,Xb,S2)
       => ( member(A,Y,S2)
         => ( ( Xb != Y )
           => aa(A,$o,aa(A,fun(A,$o),R,Xb),Y) ) ) ) ) ).

% pairwiseD
tff(fact_8175_pairwise__empty,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : pairwise(A,P,bot_bot(set(A))) ).

% pairwise_empty
tff(fact_8176_pairwise__mono,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),A3: set(A),Q: fun(A,fun(A,$o)),B3: set(A)] :
      ( pairwise(A,P,A3)
     => ( ! [X3: A,Y3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),P,X3),Y3)
           => aa(A,$o,aa(A,fun(A,$o),Q,X3),Y3) )
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B3),A3)
         => pairwise(A,Q,B3) ) ) ) ).

% pairwise_mono
tff(fact_8177_pairwise__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),S2: set(A),T3: set(A)] :
      ( pairwise(A,P,S2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S2)
       => pairwise(A,P,T3) ) ) ).

% pairwise_subset
tff(fact_8178_pairwise__singleton,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),A3: A] : pairwise(A,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ).

% pairwise_singleton
tff(fact_8179_pairwise__insert,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xb: A,S: set(A)] :
      ( pairwise(A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xb),S))
    <=> ( ! [Y4: A] :
            ( ( member(A,Y4,S)
              & ( Y4 != Xb ) )
           => ( aa(A,$o,aa(A,fun(A,$o),R2,Xb),Y4)
              & aa(A,$o,aa(A,fun(A,$o),R2,Y4),Xb) ) )
        & pairwise(A,R2,S) ) ) ).

% pairwise_insert
tff(fact_8180_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_dp(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_8181_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rq(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_8182_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_kb(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_8183_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_aao(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_4
tff(fact_8184_ATP_Olambda__5,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_yp(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_8185_ATP_Olambda__6,axiom,
    ! [A: $tType,Uu: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_ahh(set(A),$o),Uu)
    <=> ( ( Uu != bot_bot(set(A)) )
        & countable_countable(A,Uu) ) ) ).

% ATP.lambda_6
tff(fact_8186_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( aa(nat,$o,aTP_Lamp_lc(nat,$o),Uu)
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_7
tff(fact_8187_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_xj(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_8
tff(fact_8188_ATP_Olambda__9,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_aer(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),product_Pair(A,A,Uu),Uu) ).

% ATP.lambda_9
tff(fact_8189_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_cz(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_10
tff(fact_8190_ATP_Olambda__11,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_ach(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_11
tff(fact_8191_ATP_Olambda__12,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_mo(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_12
tff(fact_8192_ATP_Olambda__13,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_13
tff(fact_8193_ATP_Olambda__14,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_agd(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_14
tff(fact_8194_ATP_Olambda__15,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_aan(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Uu),Uu)) ).

% ATP.lambda_15
tff(fact_8195_ATP_Olambda__16,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_age(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_16
tff(fact_8196_ATP_Olambda__17,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_afc(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_17
tff(fact_8197_ATP_Olambda__18,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_aew(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = product_case_prod(A,C,product_prod(A,product_prod(B,C)),aTP_Lamp_aev(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_18
tff(fact_8198_ATP_Olambda__19,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_oq(real,real),Uu) = suminf(real,aTP_Lamp_hd(real,fun(nat,real),Uu)) ).

% ATP.lambda_19
tff(fact_8199_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_py(nat,set(nat)),Uu) = collect(nat,aTP_Lamp_aq(nat,fun(nat,$o),Uu)) ).

% ATP.lambda_20
tff(fact_8200_ATP_Olambda__21,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_xz(fun(A,B),set(product_prod(A,B))),Uu) = collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_xy(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_21
tff(fact_8201_ATP_Olambda__22,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(complex,complex)),aTP_Lamp_wy(real,filter(product_prod(complex,complex))),Uu) = principal(product_prod(complex,complex),collect(product_prod(complex,complex),product_case_prod(complex,complex,$o,aTP_Lamp_wx(real,fun(complex,fun(complex,$o)),Uu)))) ).

% ATP.lambda_22
tff(fact_8202_ATP_Olambda__23,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(real,real)),aTP_Lamp_ww(real,filter(product_prod(real,real))),Uu) = principal(product_prod(real,real),collect(product_prod(real,real),product_case_prod(real,real,$o,aTP_Lamp_wv(real,fun(real,fun(real,$o)),Uu)))) ).

% ATP.lambda_23
tff(fact_8203_ATP_Olambda__24,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real] : aa(real,filter(product_prod(A,A)),aTP_Lamp_wu(real,filter(product_prod(A,A))),Uu) = principal(product_prod(A,A),collect(product_prod(A,A),product_case_prod(A,A,$o,aTP_Lamp_wt(real,fun(A,fun(A,$o)),Uu)))) ) ).

% ATP.lambda_24
tff(fact_8204_ATP_Olambda__25,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_25
tff(fact_8205_ATP_Olambda__26,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_agc(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_26
tff(fact_8206_ATP_Olambda__27,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_afx(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_27
tff(fact_8207_ATP_Olambda__28,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bg(nat,fun(nat,product_prod(nat,nat))),Uu) = product_Pair(nat,nat,aa(nat,nat,suc,Uu)) ).

% ATP.lambda_28
tff(fact_8208_ATP_Olambda__29,axiom,
    ! [Uu: int] : aa(int,nat,aTP_Lamp_mq(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_29
tff(fact_8209_ATP_Olambda__30,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_zq(A,$o),Uu)
        <=> ? [N2: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N2) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2) ) ) ) ).

% ATP.lambda_30
tff(fact_8210_ATP_Olambda__31,axiom,
    ! [Uu: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aTP_Lamp_aho(fun(nat,rat),$o),Uu)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K2: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N2)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,Uu,N2)) ) ) ) ).

% ATP.lambda_31
tff(fact_8211_ATP_Olambda__32,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_aam(real,$o),Uu)
    <=> ? [I4: int,N2: nat] :
          ( ( Uu = divide_divide(real,aa(int,real,ring_1_of_int(real),I4),aa(nat,real,semiring_1_of_nat(real),N2)) )
          & ( N2 != zero_zero(nat) ) ) ) ).

% ATP.lambda_32
tff(fact_8212_ATP_Olambda__33,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_yz(product_prod(A,A),$o),Uu)
        <=> ? [X2: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X2),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4) ) ) ) ).

% ATP.lambda_33
tff(fact_8213_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_yy(product_prod(A,A),$o),Uu)
        <=> ? [X2: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X2),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X2) ) ) ) ).

% ATP.lambda_34
tff(fact_8214_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zb(product_prod(A,A),$o),Uu)
        <=> ? [X2: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X2),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4) ) ) ) ).

% ATP.lambda_35
tff(fact_8215_ATP_Olambda__36,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_za(product_prod(A,A),$o),Uu)
        <=> ? [X2: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X2),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X2) ) ) ) ).

% ATP.lambda_36
tff(fact_8216_ATP_Olambda__37,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_io(A,fun(nat,A),Uu),Uua) = $ite(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_37
tff(fact_8217_ATP_Olambda__38,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_gt(nat,fun(nat,A),Uu),Uua) = $ite(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_38
tff(fact_8218_ATP_Olambda__39,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) = $ite(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_39
tff(fact_8219_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_gs(nat,fun(nat,A),Uu),Uua) = $ite(~ 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_40
tff(fact_8220_ATP_Olambda__41,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aTP_Lamp_adx(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_41
tff(fact_8221_ATP_Olambda__42,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_xh(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_42
tff(fact_8222_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_nj(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_43
tff(fact_8223_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_aht(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_44
tff(fact_8224_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_xf(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_45
tff(fact_8225_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ahr(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_46
tff(fact_8226_ATP_Olambda__47,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_aen(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aem(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_47
tff(fact_8227_ATP_Olambda__48,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),aTP_Lamp_fk(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_48
tff(fact_8228_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gb(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_ga(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_49
tff(fact_8229_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_cw(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_50
tff(fact_8230_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hy(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_51
tff(fact_8231_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gv(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_52
tff(fact_8232_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hq(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_53
tff(fact_8233_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ip(real,fun(real,$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_54
tff(fact_8234_ATP_Olambda__55,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_aim(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Uu)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_55
tff(fact_8235_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_cv(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_56
tff(fact_8236_ATP_Olambda__57,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ade(complex,fun(real,$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_57
tff(fact_8237_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_ir(real,fun(int,$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_58
tff(fact_8238_ATP_Olambda__59,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_is(rat,fun(int,$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_59
tff(fact_8239_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hd(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_60
tff(fact_8240_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_or(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_61
tff(fact_8241_ATP_Olambda__62,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_rx(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_62
tff(fact_8242_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gu(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_63
tff(fact_8243_ATP_Olambda__64,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_aij(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),image(A,A,Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% ATP.lambda_64
tff(fact_8244_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hj(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_65
tff(fact_8245_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_he(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_66
tff(fact_8246_ATP_Olambda__67,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_kc(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_67
tff(fact_8247_ATP_Olambda__68,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_zp(set(set(A)),fun(set(A),$o),Uu),Uua)
    <=> ( member(set(A),Uua,Uu)
        & ! [X2: set(A)] :
            ( member(set(A),X2,Uu)
           => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),X2) ) ) ) ).

% ATP.lambda_68
tff(fact_8248_ATP_Olambda__69,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hx(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_69
tff(fact_8249_ATP_Olambda__70,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_kp(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_70
tff(fact_8250_ATP_Olambda__71,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_om(set(A),fun(set(A),$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_71
tff(fact_8251_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_xk(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_72
tff(fact_8252_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gx(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_73
tff(fact_8253_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gw(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_74
tff(fact_8254_ATP_Olambda__75,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_aaj(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),product_Pair(B,A,Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_75
tff(fact_8255_ATP_Olambda__76,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_adt(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_76
tff(fact_8256_ATP_Olambda__77,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_cs(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,power_power(complex,Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_77
tff(fact_8257_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aw(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,power_power(A,Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_78
tff(fact_8258_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_yo(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_79
tff(fact_8259_ATP_Olambda__80,axiom,
    ! [A: $tType,Uu: fun(set(A),$o),Uua: set(A)] :
      ( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_yx(fun(set(A),$o),fun(set(A),$o)),Uu),Uua)
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A8: set(A),A7: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A7),A8) )
            & aa(set(A),$o,Uu,A8) ) ) ) ).

% ATP.lambda_80
tff(fact_8260_ATP_Olambda__81,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_ahd(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))),Uu) ).

% ATP.lambda_81
tff(fact_8261_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hs(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_82
tff(fact_8262_ATP_Olambda__83,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aax(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_83
tff(fact_8263_ATP_Olambda__84,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aaz(nat,fun(nat,$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_84
tff(fact_8264_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ez(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),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_85
tff(fact_8265_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),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_86
tff(fact_8266_ATP_Olambda__87,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,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_87
tff(fact_8267_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ck(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_88
tff(fact_8268_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_iz(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_89
tff(fact_8269_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ix(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_90
tff(fact_8270_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dx(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_91
tff(fact_8271_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_da(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_92
tff(fact_8272_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bq(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_93
tff(fact_8273_ATP_Olambda__94,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fm(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_94
tff(fact_8274_ATP_Olambda__95,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_95
tff(fact_8275_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_96
tff(fact_8276_ATP_Olambda__97,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,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_97
tff(fact_8277_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_de(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_98
tff(fact_8278_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aia(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y4: A] :
                ( aa(A,$o,Uu,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y4) ) ) ) ) ).

% ATP.lambda_99
tff(fact_8279_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_vz(fun(A,$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_100
tff(fact_8280_ATP_Olambda__101,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),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_101
tff(fact_8281_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fs(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_102
tff(fact_8282_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,A,aTP_Lamp_aib(fun(A,real),fun(A,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(A,real,Uu,Uua)),Uua) ) ).

% ATP.lambda_103
tff(fact_8283_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahp(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_104
tff(fact_8284_ATP_Olambda__105,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_vb(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_105
tff(fact_8285_ATP_Olambda__106,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_aec(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_106
tff(fact_8286_ATP_Olambda__107,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_lt(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_107
tff(fact_8287_ATP_Olambda__108,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ch(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_108
tff(fact_8288_ATP_Olambda__109,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ep(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_109
tff(fact_8289_ATP_Olambda__110,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ry(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_rx(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_110
tff(fact_8290_ATP_Olambda__111,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_rx(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_111
tff(fact_8291_ATP_Olambda__112,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_aep(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_acd(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_112
tff(fact_8292_ATP_Olambda__113,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_xs(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_113
tff(fact_8293_ATP_Olambda__114,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_td(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_114
tff(fact_8294_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ih(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_115
tff(fact_8295_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ij(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_116
tff(fact_8296_ATP_Olambda__117,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,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_117
tff(fact_8297_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dr(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_118
tff(fact_8298_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_nl(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_119
tff(fact_8299_ATP_Olambda__120,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_120
tff(fact_8300_ATP_Olambda__121,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_121
tff(fact_8301_ATP_Olambda__122,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ic(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_122
tff(fact_8302_ATP_Olambda__123,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_im(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_123
tff(fact_8303_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_wn(A,fun(set(A),$o),Uu),Uua)
        <=> ( topolo1002775350975398744n_open(A,Uua)
            & member(A,Uu,Uua) ) ) ) ).

% ATP.lambda_124
tff(fact_8304_ATP_Olambda__125,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_abg(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_125
tff(fact_8305_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kt(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_126
tff(fact_8306_ATP_Olambda__127,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_aeo(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_127
tff(fact_8307_ATP_Olambda__128,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_abx(list(A),fun(list(A),$o)),Uu),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_128
tff(fact_8308_ATP_Olambda__129,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_ain(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_129
tff(fact_8309_ATP_Olambda__130,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ahb(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_130
tff(fact_8310_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hp(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_131
tff(fact_8311_ATP_Olambda__132,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_aeq(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),product_Pair(A,B,Uua)),Uu) ).

% ATP.lambda_132
tff(fact_8312_ATP_Olambda__133,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_acj(set(nat),fun(nat,$o),Uu),Uua)
    <=> member(nat,aa(nat,nat,suc,Uua),Uu) ) ).

% ATP.lambda_133
tff(fact_8313_ATP_Olambda__134,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_bs(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),product_Pair(A,A,Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_134
tff(fact_8314_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_do(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_135
tff(fact_8315_ATP_Olambda__136,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_aad(A,fun(set(set(A)),set(set(A)))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),Uua),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu)),Uua)) ).

% ATP.lambda_136
tff(fact_8316_ATP_Olambda__137,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_aih(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = image(B,A,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_137
tff(fact_8317_ATP_Olambda__138,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_aix(fun(A,B),fun(B,set(A)),Uu),Uua) = vimage(A,B,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_138
tff(fact_8318_ATP_Olambda__139,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_iq(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_139
tff(fact_8319_ATP_Olambda__140,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_140
tff(fact_8320_ATP_Olambda__141,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_aed(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),product_Pair(nat,A,Uua),aa(nat,A,Uu,Uua)) ).

% ATP.lambda_141
tff(fact_8321_ATP_Olambda__142,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_aez(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_142
tff(fact_8322_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ka(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_143
tff(fact_8323_ATP_Olambda__144,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_afa(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).

% ATP.lambda_144
tff(fact_8324_ATP_Olambda__145,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_145
tff(fact_8325_ATP_Olambda__146,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aay(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_146
tff(fact_8326_ATP_Olambda__147,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_147
tff(fact_8327_ATP_Olambda__148,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_148
tff(fact_8328_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hw(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_149
tff(fact_8329_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jf(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_150
tff(fact_8330_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ex(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_151
tff(fact_8331_ATP_Olambda__152,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afn(list(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_152
tff(fact_8332_ATP_Olambda__153,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_agr(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_153
tff(fact_8333_ATP_Olambda__154,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aiq(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua = last(A,Uu) ) ) ).

% ATP.lambda_154
tff(fact_8334_ATP_Olambda__155,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_155
tff(fact_8335_ATP_Olambda__156,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_156
tff(fact_8336_ATP_Olambda__157,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aaa(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_157
tff(fact_8337_ATP_Olambda__158,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tx(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_158
tff(fact_8338_ATP_Olambda__159,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_159
tff(fact_8339_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ni(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_160
tff(fact_8340_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_lo(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_161
tff(fact_8341_ATP_Olambda__162,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_xt(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_162
tff(fact_8342_ATP_Olambda__163,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_163
tff(fact_8343_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aab(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_164
tff(fact_8344_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ty(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_165
tff(fact_8345_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_db(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_166
tff(fact_8346_ATP_Olambda__167,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_167
tff(fact_8347_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mf(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_168
tff(fact_8348_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_lf(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_169
tff(fact_8349_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ln(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_170
tff(fact_8350_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_nx(nat,fun(real,real),Uu),Uua) = aa(nat,real,power_power(real,Uua),Uu) ).

% ATP.lambda_171
tff(fact_8351_ATP_Olambda__172,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_mu(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_172
tff(fact_8352_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aeu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_173
tff(fact_8353_ATP_Olambda__174,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_mx(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_174
tff(fact_8354_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_xu(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_8355_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_lm(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_176
tff(fact_8356_ATP_Olambda__177,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_nz(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_177
tff(fact_8357_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aq(nat,fun(nat,$o),Uu),Uua)
    <=> dvd_dvd(nat,Uua,Uu) ) ).

% ATP.lambda_178
tff(fact_8358_ATP_Olambda__179,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_aj(int,fun(int,$o),Uu),Uua)
    <=> dvd_dvd(int,Uua,Uu) ) ).

% ATP.lambda_179
tff(fact_8359_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_al(A,fun(A,$o),Uu),Uua)
        <=> dvd_dvd(A,Uua,Uu) ) ) ).

% ATP.lambda_180
tff(fact_8360_ATP_Olambda__181,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ks(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uua),Uu) ).

% ATP.lambda_181
tff(fact_8361_ATP_Olambda__182,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_aey(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uu) ).

% ATP.lambda_182
tff(fact_8362_ATP_Olambda__183,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_afb(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),product_Pair(nat,A,Uua),Uu) ).

% ATP.lambda_183
tff(fact_8363_ATP_Olambda__184,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_acd(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_184
tff(fact_8364_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_ye(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_185
tff(fact_8365_ATP_Olambda__186,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_186
tff(fact_8366_ATP_Olambda__187,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_xl(set(B),fun(B,$o),Uu),Uua)
    <=> member(B,Uua,Uu) ) ).

% ATP.lambda_187
tff(fact_8367_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_xp(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_188
tff(fact_8368_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahv(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_189
tff(fact_8369_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahz(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_190
tff(fact_8370_ATP_Olambda__191,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_191
tff(fact_8371_ATP_Olambda__192,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_aaw(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_192
tff(fact_8372_ATP_Olambda__193,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_aco(list(A),fun(nat,list(A)),Uu),Uua) = drop(A,Uua,Uu) ).

% ATP.lambda_193
tff(fact_8373_ATP_Olambda__194,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_194
tff(fact_8374_ATP_Olambda__195,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_bm(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_195
tff(fact_8375_ATP_Olambda__196,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aes(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_196
tff(fact_8376_ATP_Olambda__197,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_aet(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_197
tff(fact_8377_ATP_Olambda__198,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_vg(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_198
tff(fact_8378_ATP_Olambda__199,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_va(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_199
tff(fact_8379_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aeb(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_200
tff(fact_8380_ATP_Olambda__201,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_dm(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_201
tff(fact_8381_ATP_Olambda__202,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_dl(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_202
tff(fact_8382_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_to(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,divide_divide(A,one_one(A),Uua)) ) ).

% ATP.lambda_203
tff(fact_8383_ATP_Olambda__204,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ahw(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_204
tff(fact_8384_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_205
tff(fact_8385_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_el(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_206
tff(fact_8386_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_eh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_207
tff(fact_8387_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cf(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_208
tff(fact_8388_ATP_Olambda__209,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ael(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_209
tff(fact_8389_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_abt(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_210
tff(fact_8390_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_abi(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_211
tff(fact_8391_ATP_Olambda__212,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_abj(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_212
tff(fact_8392_ATP_Olambda__213,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_abk(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_213
tff(fact_8393_ATP_Olambda__214,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_acr(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_acq(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_214
tff(fact_8394_ATP_Olambda__215,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_lb(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_la(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_215
tff(fact_8395_ATP_Olambda__216,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) = 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_216
tff(fact_8396_ATP_Olambda__217,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_kx(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_kw(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_217
tff(fact_8397_ATP_Olambda__218,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_kv(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_ku(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_218
tff(fact_8398_ATP_Olambda__219,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_kr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_kq(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_219
tff(fact_8399_ATP_Olambda__220,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_oh(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_og(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_220
tff(fact_8400_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_ob(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_hl(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_221
tff(fact_8401_ATP_Olambda__222,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A] : aa(A,set(A),aTP_Lamp_tm(real,fun(A,set(A)),Uu),Uua) = collect(A,aa(A,fun(A,$o),aTP_Lamp_tl(real,fun(A,fun(A,$o)),Uu),Uua)) ) ).

% ATP.lambda_222
tff(fact_8402_ATP_Olambda__223,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_adb(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_223
tff(fact_8403_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_pj(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_224
tff(fact_8404_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aic(fun(A,real),fun(A,$o),Uu),Uua)
        <=> ( aa(A,real,Uu,Uua) != zero_zero(real) ) ) ) ).

% ATP.lambda_225
tff(fact_8405_ATP_Olambda__226,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_afy(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_226
tff(fact_8406_ATP_Olambda__227,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_adu(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_adt(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_227
tff(fact_8407_ATP_Olambda__228,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_nh(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_228
tff(fact_8408_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ht(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_229
tff(fact_8409_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_dq(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_230
tff(fact_8410_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afo(list(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_231
tff(fact_8411_ATP_Olambda__232,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aip(list(A),fun(A,$o),Uu),Uua)
    <=> ( Uua != last(A,Uu) ) ) ).

% ATP.lambda_232
tff(fact_8412_ATP_Olambda__233,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ady(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_233
tff(fact_8413_ATP_Olambda__234,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_234
tff(fact_8414_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nu(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_235
tff(fact_8415_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_wl(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu,Uua)) ) ).

% ATP.lambda_236
tff(fact_8416_ATP_Olambda__237,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_wk(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu)) ) ).

% ATP.lambda_237
tff(fact_8417_ATP_Olambda__238,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_aak(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),product_Pair(B,A,Uu),Uua)) ).

% ATP.lambda_238
tff(fact_8418_ATP_Olambda__239,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_adv(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Uu),Uua)) ).

% ATP.lambda_239
tff(fact_8419_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abv(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uu),Uua)) ).

% ATP.lambda_240
tff(fact_8420_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uua),Uu)) ).

% ATP.lambda_241
tff(fact_8421_ATP_Olambda__242,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu),Uua)) ).

% ATP.lambda_242
tff(fact_8422_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jy(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu)) ).

% ATP.lambda_243
tff(fact_8423_ATP_Olambda__244,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_bc(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_244
tff(fact_8424_ATP_Olambda__245,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_afh(A,fun(A,$o),Uu),Uua)
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_245
tff(fact_8425_ATP_Olambda__246,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afm(A,fun(A,$o)),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_246
tff(fact_8426_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ahq(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_247
tff(fact_8427_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_fp(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_248
tff(fact_8428_ATP_Olambda__249,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_qq(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_249
tff(fact_8429_ATP_Olambda__250,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: A] : aa(A,B,aTP_Lamp_kf(fun(A,$o),fun(A,B),Uu),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_8430_ATP_Olambda__251,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,set(B),aTP_Lamp_ado(fun(A,B),fun(A,set(B)),Uu),Uua) = aa(B,set(B),set_ord_atMost(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_8431_ATP_Olambda__252,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_252
tff(fact_8432_ATP_Olambda__253,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ph(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_8433_ATP_Olambda__254,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ur(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_254
tff(fact_8434_ATP_Olambda__255,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_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_255
tff(fact_8435_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_nt(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_256
tff(fact_8436_ATP_Olambda__257,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_rn(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_257
tff(fact_8437_ATP_Olambda__258,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_258
tff(fact_8438_ATP_Olambda__259,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_qp(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_259
tff(fact_8439_ATP_Olambda__260,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_ds(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_260
tff(fact_8440_ATP_Olambda__261,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_cp(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_261
tff(fact_8441_ATP_Olambda__262,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_cq(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_262
tff(fact_8442_ATP_Olambda__263,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pk(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_263
tff(fact_8443_ATP_Olambda__264,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_fz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_264
tff(fact_8444_ATP_Olambda__265,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ed(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_265
tff(fact_8445_ATP_Olambda__266,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_266
tff(fact_8446_ATP_Olambda__267,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_bp(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_267
tff(fact_8447_ATP_Olambda__268,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fa(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_268
tff(fact_8448_ATP_Olambda__269,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ri(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_269
tff(fact_8449_ATP_Olambda__270,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_os(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_270
tff(fact_8450_ATP_Olambda__271,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tk(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_271
tff(fact_8451_ATP_Olambda__272,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qi(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_272
tff(fact_8452_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_273
tff(fact_8453_ATP_Olambda__274,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_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_274
tff(fact_8454_ATP_Olambda__275,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ro(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8455_ATP_Olambda__276,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_rb(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8456_ATP_Olambda__277,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_hi(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_277
tff(fact_8457_ATP_Olambda__278,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_hh(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8458_ATP_Olambda__279,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_vt(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_279
tff(fact_8459_ATP_Olambda__280,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_rs(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_280
tff(fact_8460_ATP_Olambda__281,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_oe(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_281
tff(fact_8461_ATP_Olambda__282,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_qk(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_282
tff(fact_8462_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ql(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_283
tff(fact_8463_ATP_Olambda__284,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_ib(fun(A,B),fun(A,B),Uu),Uua) = exp(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_284
tff(fact_8464_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qj(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_8465_ATP_Olambda__286,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_aei(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = product_Pair(A,B,aa(C,A,Uu,Uua)) ).

% ATP.lambda_286
tff(fact_8466_ATP_Olambda__287,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_wi(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_287
tff(fact_8467_ATP_Olambda__288,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_wj(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_288
tff(fact_8468_ATP_Olambda__289,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_na(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_289
tff(fact_8469_ATP_Olambda__290,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_rv(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8470_ATP_Olambda__291,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_yb(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_291
tff(fact_8471_ATP_Olambda__292,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ou(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_8472_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_aae(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_293
tff(fact_8473_ATP_Olambda__294,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_jo(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_294
tff(fact_8474_ATP_Olambda__295,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_zt(fun(B,$o),fun(B,$o),Uu),Uua)
    <=> ~ aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_295
tff(fact_8475_ATP_Olambda__296,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_be(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_296
tff(fact_8476_ATP_Olambda__297,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_adz(nat,fun(nat,set(nat)),Uu),Uua) = collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Uu)) ).

% ATP.lambda_297
tff(fact_8477_ATP_Olambda__298,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_abf(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),collect(set(A),aa(set(A),fun(set(A),$o),aTP_Lamp_abe(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua))) ).

% ATP.lambda_298
tff(fact_8478_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real] : aa(real,filter(A),aTP_Lamp_wh(A,fun(real,filter(A)),Uu),Uua) = principal(A,collect(A,aa(real,fun(A,$o),aTP_Lamp_wg(A,fun(real,fun(A,$o)),Uu),Uua))) ) ).

% ATP.lambda_299
tff(fact_8479_ATP_Olambda__300,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_zg(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_300
tff(fact_8480_ATP_Olambda__301,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_zl(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) )
              & ! [X2: set(A)] :
                  ( member(set(A),X2,Uu)
                 => member(A,aa(set(A),A,F7,X2),X2) ) ) ) ) ).

% ATP.lambda_301
tff(fact_8481_ATP_Olambda__302,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_zk(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) )
              & ! [X2: set(A)] :
                  ( member(set(A),X2,Uu)
                 => member(A,aa(set(A),A,F7,X2),X2) ) ) ) ) ).

% ATP.lambda_302
tff(fact_8482_ATP_Olambda__303,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_zm(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) )
              & ! [X2: set(A)] :
                  ( member(set(A),X2,Uu)
                 => member(A,aa(set(A),A,F7,X2),X2) ) ) ) ) ).

% ATP.lambda_303
tff(fact_8483_ATP_Olambda__304,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_zo(set(A),fun(set(A),$o),Uu),Uua)
    <=> ? [B11: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B11) )
          & member(set(A),Uu,pow2(A,B11)) ) ) ).

% ATP.lambda_304
tff(fact_8484_ATP_Olambda__305,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_zh(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X2: A] :
              ( member(A,X2,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X2) ) ) ) ).

% ATP.lambda_305
tff(fact_8485_ATP_Olambda__306,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_zf(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X2: A] :
              ( member(A,X2,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X2) ) ) ) ).

% ATP.lambda_306
tff(fact_8486_ATP_Olambda__307,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_zi(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X2: A] :
              ( member(A,X2,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Uua) ) ) ) ).

% ATP.lambda_307
tff(fact_8487_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ze(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X2: A] :
              ( member(A,X2,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Uua) ) ) ) ).

% ATP.lambda_308
tff(fact_8488_ATP_Olambda__309,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_vs(fun(A,$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_309
tff(fact_8489_ATP_Olambda__310,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_zc(fun(B,A),fun(A,$o),Uu),Uua)
    <=> ? [X2: B] : Uua = aa(B,A,Uu,X2) ) ).

% ATP.lambda_310
tff(fact_8490_ATP_Olambda__311,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_agz(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu),Uua)
    <=> ? [X8: set(A),Y6: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),X8),Y6) )
          & ( X8 != bot_bot(set(A)) )
          & ! [X2: A] :
              ( member(A,X2,Y6)
             => ? [Xa3: A] :
                  ( member(A,Xa3,X8)
                  & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X2),Uu) ) ) ) ) ).

% ATP.lambda_311
tff(fact_8491_ATP_Olambda__312,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_agq(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_312
tff(fact_8492_ATP_Olambda__313,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_adk(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_313
tff(fact_8493_ATP_Olambda__314,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(set(A),set(A)),aa(A,fun(list(A),fun(set(A),set(A))),aTP_Lamp_ail(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert2(A),Uu) ).

% ATP.lambda_314
tff(fact_8494_ATP_Olambda__315,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_ahc(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_315
tff(fact_8495_ATP_Olambda__316,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_dk(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_316
tff(fact_8496_ATP_Olambda__317,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bf(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_317
tff(fact_8497_ATP_Olambda__318,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_bb(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),product_Pair(A,A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),product_Pair(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_318
tff(fact_8498_ATP_Olambda__319,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_acx(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_319
tff(fact_8499_ATP_Olambda__320,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_eo(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_320
tff(fact_8500_ATP_Olambda__321,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_kh(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_321
tff(fact_8501_ATP_Olambda__322,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_ki(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_322
tff(fact_8502_ATP_Olambda__323,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_bn(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_323
tff(fact_8503_ATP_Olambda__324,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_324
tff(fact_8504_ATP_Olambda__325,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_dy(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_325
tff(fact_8505_ATP_Olambda__326,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_326
tff(fact_8506_ATP_Olambda__327,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_327
tff(fact_8507_ATP_Olambda__328,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ais(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_328
tff(fact_8508_ATP_Olambda__329,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_air(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).

% ATP.lambda_329
tff(fact_8509_ATP_Olambda__330,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_aag(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),Uub),Uub) ).

% ATP.lambda_330
tff(fact_8510_ATP_Olambda__331,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_en(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_331
tff(fact_8511_ATP_Olambda__332,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
          aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_afi(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_332
tff(fact_8512_ATP_Olambda__333,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,$o),fun(A,B),aTP_Lamp_cd(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_333
tff(fact_8513_ATP_Olambda__334,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,$o),fun(A,B),aTP_Lamp_eg(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_334
tff(fact_8514_ATP_Olambda__335,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,$o),fun(B,fun(A,A)),aTP_Lamp_afz(fun(B,fun(A,A)),fun(fun(B,$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_335
tff(fact_8515_ATP_Olambda__336,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_agf(fun(B,A),fun(fun(B,$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_336
tff(fact_8516_ATP_Olambda__337,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_adw(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_adv(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_337
tff(fact_8517_ATP_Olambda__338,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_aap(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_aak(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_338
tff(fact_8518_ATP_Olambda__339,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ya(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_339
tff(fact_8519_ATP_Olambda__340,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_fk(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_340
tff(fact_8520_ATP_Olambda__341,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_ga(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_341
tff(fact_8521_ATP_Olambda__342,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_fw(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_342
tff(fact_8522_ATP_Olambda__343,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_ft(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_343
tff(fact_8523_ATP_Olambda__344,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_ys(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_344
tff(fact_8524_ATP_Olambda__345,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_qm(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_345
tff(fact_8525_ATP_Olambda__346,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_ud(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_346
tff(fact_8526_ATP_Olambda__347,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_pq(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_347
tff(fact_8527_ATP_Olambda__348,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_eb(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_348
tff(fact_8528_ATP_Olambda__349,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_by(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_349
tff(fact_8529_ATP_Olambda__350,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agn(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_350
tff(fact_8530_ATP_Olambda__351,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_351
tff(fact_8531_ATP_Olambda__352,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_ie(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_id(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_352
tff(fact_8532_ATP_Olambda__353,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_ia(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_hz(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_353
tff(fact_8533_ATP_Olambda__354,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_es(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_354
tff(fact_8534_ATP_Olambda__355,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_aio(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        product_Pair(code_integer,$o,
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_355
tff(fact_8535_ATP_Olambda__356,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_fx(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_fw(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_356
tff(fact_8536_ATP_Olambda__357,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_fu(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_ft(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_357
tff(fact_8537_ATP_Olambda__358,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_aje(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_ajd(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_358
tff(fact_8538_ATP_Olambda__359,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_pr(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_pq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_359
tff(fact_8539_ATP_Olambda__360,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_ue(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_ud(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_360
tff(fact_8540_ATP_Olambda__361,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_oi(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_361
tff(fact_8541_ATP_Olambda__362,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_oj(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_362
tff(fact_8542_ATP_Olambda__363,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_hu(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_363
tff(fact_8543_ATP_Olambda__364,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_gf(fun(nat,A),fun(nat,fun(A,$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_fr(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_364
tff(fact_8544_ATP_Olambda__365,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_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,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua)),Uub) ) ).

% ATP.lambda_365
tff(fact_8545_ATP_Olambda__366,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_rk(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_366
tff(fact_8546_ATP_Olambda__367,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_br(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_367
tff(fact_8547_ATP_Olambda__368,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_rl(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_368
tff(fact_8548_ATP_Olambda__369,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_of(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_369
tff(fact_8549_ATP_Olambda__370,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_hv(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_370
tff(fact_8550_ATP_Olambda__371,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_371
tff(fact_8551_ATP_Olambda__372,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_hg(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_372
tff(fact_8552_ATP_Olambda__373,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),$o),aTP_Lamp_abs(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Uua),Uub),lex(A,Uu)) ) ) ) ).

% ATP.lambda_373
tff(fact_8553_ATP_Olambda__374,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),$o),aTP_Lamp_acn(set(product_prod(A,A)),fun(list(A),fun(list(A),$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),X2: A,Y4: A,Xs5: list(A),Ys6: 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),X2),Xs5)) )
            & ( 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),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X2),Y4),Uu) ) ) ) ).

% ATP.lambda_374
tff(fact_8554_ATP_Olambda__375,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_375
tff(fact_8555_ATP_Olambda__376,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_js(nat,fun(set(A),fun(list(A),$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_376
tff(fact_8556_ATP_Olambda__377,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_jr(set(A),fun(nat,fun(list(A),$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_377
tff(fact_8557_ATP_Olambda__378,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_adg(set(A),fun(nat,fun(list(A),$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_378
tff(fact_8558_ATP_Olambda__379,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_jt(nat,fun(list(A),fun(list(A),$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_379
tff(fact_8559_ATP_Olambda__380,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(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_380
tff(fact_8560_ATP_Olambda__381,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_at(set(A),fun(nat,fun(list(A),$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_381
tff(fact_8561_ATP_Olambda__382,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_382
tff(fact_8562_ATP_Olambda__383,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)),$o),aTP_Lamp_acb(set(A),fun(set(B),fun(fun(A,option(B)),$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_383
tff(fact_8563_ATP_Olambda__384,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_oa(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_384
tff(fact_8564_ATP_Olambda__385,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jm(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_385
tff(fact_8565_ATP_Olambda__386,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_jz(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_386
tff(fact_8566_ATP_Olambda__387,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_abd(list(A),fun(set(nat),fun(nat,$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_387
tff(fact_8567_ATP_Olambda__388,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_afp(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_388
tff(fact_8568_ATP_Olambda__389,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_agl(list(A),fun(fun(A,$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_389
tff(fact_8569_ATP_Olambda__390,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_aff(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_390
tff(fact_8570_ATP_Olambda__391,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_je(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_391
tff(fact_8571_ATP_Olambda__392,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_aci(list(A),fun(set(nat),fun(nat,$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_392
tff(fact_8572_ATP_Olambda__393,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uub) ) ) ).

% ATP.lambda_393
tff(fact_8573_ATP_Olambda__394,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_ahu(set(A),fun(nat,fun(A,$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_394
tff(fact_8574_ATP_Olambda__395,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,$o),aTP_Lamp_adf(set(A),fun(fun(A,B),fun(A,$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_395
tff(fact_8575_ATP_Olambda__396,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_agx(set(nat),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & member(nat,aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_agw(set(nat),fun(nat,fun(nat,$o)),Uu),Uub))),Uua) ) ) ).

% ATP.lambda_396
tff(fact_8576_ATP_Olambda__397,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_jk(set(A),fun(nat,fun(set(A),$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_397
tff(fact_8577_ATP_Olambda__398,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jn(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_398
tff(fact_8578_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hf(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_399
tff(fact_8579_ATP_Olambda__400,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_ny(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_400
tff(fact_8580_ATP_Olambda__401,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_hm(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_401
tff(fact_8581_ATP_Olambda__402,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_ahn(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))),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_402
tff(fact_8582_ATP_Olambda__403,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gy(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_403
tff(fact_8583_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_zw(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_404
tff(fact_8584_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bt(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_405
tff(fact_8585_ATP_Olambda__406,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bh(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_406
tff(fact_8586_ATP_Olambda__407,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_407
tff(fact_8587_ATP_Olambda__408,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bi(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_408
tff(fact_8588_ATP_Olambda__409,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_409
tff(fact_8589_ATP_Olambda__410,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_410
tff(fact_8590_ATP_Olambda__411,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_411
tff(fact_8591_ATP_Olambda__412,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_ahl(set(A),fun(set(A),fun(set(A),$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_412
tff(fact_8592_ATP_Olambda__413,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_agw(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_413
tff(fact_8593_ATP_Olambda__414,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_kg(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & member(A,Uub,Uua) ) ) ).

% ATP.lambda_414
tff(fact_8594_ATP_Olambda__415,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_vv(set(A),fun(fun(A,real),fun(A,$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_415
tff(fact_8595_ATP_Olambda__416,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_afd(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_416
tff(fact_8596_ATP_Olambda__417,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_417
tff(fact_8597_ATP_Olambda__418,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahx(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_418
tff(fact_8598_ATP_Olambda__419,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ak(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_419
tff(fact_8599_ATP_Olambda__420,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aah(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_420
tff(fact_8600_ATP_Olambda__421,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ca(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_421
tff(fact_8601_ATP_Olambda__422,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cb(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_422
tff(fact_8602_ATP_Olambda__423,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,$o),aTP_Lamp_adn(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_423
tff(fact_8603_ATP_Olambda__424,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,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_424
tff(fact_8604_ATP_Olambda__425,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,$o),aTP_Lamp_it(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_425
tff(fact_8605_ATP_Olambda__426,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,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_426
tff(fact_8606_ATP_Olambda__427,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,$o),aTP_Lamp_iu(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_427
tff(fact_8607_ATP_Olambda__428,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,$o),aTP_Lamp_le(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_428
tff(fact_8608_ATP_Olambda__429,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,$o),aTP_Lamp_jj(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ~ dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2)),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_429
tff(fact_8609_ATP_Olambda__430,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bd(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_430
tff(fact_8610_ATP_Olambda__431,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_rt(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_431
tff(fact_8611_ATP_Olambda__432,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_fj(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_432
tff(fact_8612_ATP_Olambda__433,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_tg(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_433
tff(fact_8613_ATP_Olambda__434,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_wx(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_434
tff(fact_8614_ATP_Olambda__435,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_wv(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_435
tff(fact_8615_ATP_Olambda__436,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_wt(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_436
tff(fact_8616_ATP_Olambda__437,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tl(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_437
tff(fact_8617_ATP_Olambda__438,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_wg(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_438
tff(fact_8618_ATP_Olambda__439,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_fn(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_439
tff(fact_8619_ATP_Olambda__440,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mj(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_440
tff(fact_8620_ATP_Olambda__441,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mh(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_441
tff(fact_8621_ATP_Olambda__442,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mi(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_442
tff(fact_8622_ATP_Olambda__443,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mg(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_443
tff(fact_8623_ATP_Olambda__444,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aig(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,Uua),Uub),Uu) ) ).

% ATP.lambda_444
tff(fact_8624_ATP_Olambda__445,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aby(set(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uub),Uu) ) ).

% ATP.lambda_445
tff(fact_8625_ATP_Olambda__446,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aem(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_446
tff(fact_8626_ATP_Olambda__447,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_av(nat,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_447
tff(fact_8627_ATP_Olambda__448,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_jp(complex,fun(nat,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_448
tff(fact_8628_ATP_Olambda__449,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_yn(A,fun(A,fun(A,$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_449
tff(fact_8629_ATP_Olambda__450,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_gc(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_450
tff(fact_8630_ATP_Olambda__451,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_og(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_451
tff(fact_8631_ATP_Olambda__452,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_gd(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_452
tff(fact_8632_ATP_Olambda__453,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gi(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_453
tff(fact_8633_ATP_Olambda__454,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_pz(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_454
tff(fact_8634_ATP_Olambda__455,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_fr(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_455
tff(fact_8635_ATP_Olambda__456,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(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_456
tff(fact_8636_ATP_Olambda__457,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_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_457
tff(fact_8637_ATP_Olambda__458,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_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,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_458
tff(fact_8638_ATP_Olambda__459,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_fy(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_459
tff(fact_8639_ATP_Olambda__460,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fb(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_460
tff(fact_8640_ATP_Olambda__461,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_461
tff(fact_8641_ATP_Olambda__462,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_ow(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_462
tff(fact_8642_ATP_Olambda__463,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_adr(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_463
tff(fact_8643_ATP_Olambda__464,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_adq(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_464
tff(fact_8644_ATP_Olambda__465,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_agh(fun(B,A),fun(B,fun(B,$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_465
tff(fact_8645_ATP_Olambda__466,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,$o),aTP_Lamp_xo(fun(A,B),fun(fun(A,B),fun(A,$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_466
tff(fact_8646_ATP_Olambda__467,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,$o),aTP_Lamp_uf(fun(A,B),fun(fun(A,B),fun(A,$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_467
tff(fact_8647_ATP_Olambda__468,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,$o),aTP_Lamp_uk(fun(A,B),fun(fun(A,B),fun(A,$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_468
tff(fact_8648_ATP_Olambda__469,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,$o),aTP_Lamp_ux(fun(A,B),fun(fun(A,B),fun(A,$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_469
tff(fact_8649_ATP_Olambda__470,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_po(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_470
tff(fact_8650_ATP_Olambda__471,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_pf(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_471
tff(fact_8651_ATP_Olambda__472,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_vm(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_472
tff(fact_8652_ATP_Olambda__473,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_ns(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_473
tff(fact_8653_ATP_Olambda__474,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_rm(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_474
tff(fact_8654_ATP_Olambda__475,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_vi(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_475
tff(fact_8655_ATP_Olambda__476,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_tr(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_476
tff(fact_8656_ATP_Olambda__477,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_qz(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_477
tff(fact_8657_ATP_Olambda__478,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,$o),aTP_Lamp_vq(fun(A,B),fun(fun(A,B),fun(A,$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_478
tff(fact_8658_ATP_Olambda__479,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_ahm(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_479
tff(fact_8659_ATP_Olambda__480,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_ji(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_480
tff(fact_8660_ATP_Olambda__481,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_jh(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_481
tff(fact_8661_ATP_Olambda__482,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_pb(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_482
tff(fact_8662_ATP_Olambda__483,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_no(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_483
tff(fact_8663_ATP_Olambda__484,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_484
tff(fact_8664_ATP_Olambda__485,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_qr(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_485
tff(fact_8665_ATP_Olambda__486,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_tq(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_486
tff(fact_8666_ATP_Olambda__487,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_ld(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_487
tff(fact_8667_ATP_Olambda__488,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_sy(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_488
tff(fact_8668_ATP_Olambda__489,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_hk(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_489
tff(fact_8669_ATP_Olambda__490,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_qx(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_490
tff(fact_8670_ATP_Olambda__491,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_eu(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_491
tff(fact_8671_ATP_Olambda__492,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_dd(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_492
tff(fact_8672_ATP_Olambda__493,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_ce(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_493
tff(fact_8673_ATP_Olambda__494,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_qw(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_494
tff(fact_8674_ATP_Olambda__495,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_zy(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_495
tff(fact_8675_ATP_Olambda__496,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_wp(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_496
tff(fact_8676_ATP_Olambda__497,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_dz(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_497
tff(fact_8677_ATP_Olambda__498,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_bv(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_498
tff(fact_8678_ATP_Olambda__499,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_pa(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_499
tff(fact_8679_ATP_Olambda__500,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_vl(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_500
tff(fact_8680_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_nn(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_501
tff(fact_8681_ATP_Olambda__502,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_qo(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_502
tff(fact_8682_ATP_Olambda__503,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_tp(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_503
tff(fact_8683_ATP_Olambda__504,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_504
tff(fact_8684_ATP_Olambda__505,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_iv(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_505
tff(fact_8685_ATP_Olambda__506,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_od(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_506
tff(fact_8686_ATP_Olambda__507,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_pu(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_507
tff(fact_8687_ATP_Olambda__508,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_vw(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_508
tff(fact_8688_ATP_Olambda__509,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_vh(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_509
tff(fact_8689_ATP_Olambda__510,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_vx(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_510
tff(fact_8690_ATP_Olambda__511,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_rw(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_511
tff(fact_8691_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_rh(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_512
tff(fact_8692_ATP_Olambda__513,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_aiy(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),product_Pair(A,B,aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_513
tff(fact_8693_ATP_Olambda__514,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_zx(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_514
tff(fact_8694_ATP_Olambda__515,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ajg(fun(nat,set(A)),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> disjnt(A,aa(nat,set(A),Uu,Uua),aa(nat,set(A),Uu,Uub)) ) ).

% ATP.lambda_515
tff(fact_8695_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajf(fun(A,set(B)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> disjnt(B,aa(A,set(B),Uu,Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_516
tff(fact_8696_ATP_Olambda__517,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_aat(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_517
tff(fact_8697_ATP_Olambda__518,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_518
tff(fact_8698_ATP_Olambda__519,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_519
tff(fact_8699_ATP_Olambda__520,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_afe(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_520
tff(fact_8700_ATP_Olambda__521,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afj(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_521
tff(fact_8701_ATP_Olambda__522,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_agm(A,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_522
tff(fact_8702_ATP_Olambda__523,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ke(fun(A,B),fun(fun(A,$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_523
tff(fact_8703_ATP_Olambda__524,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,$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_524
tff(fact_8704_ATP_Olambda__525,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_wa(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_525
tff(fact_8705_ATP_Olambda__526,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_nv(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_526
tff(fact_8706_ATP_Olambda__527,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_adp(fun(A,B),fun(B,fun(A,set(B))),Uu),Uua),Uub) = set_or3652927894154168847AtMost(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_527
tff(fact_8707_ATP_Olambda__528,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_qn(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_528
tff(fact_8708_ATP_Olambda__529,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( aa(nat,$o,aa(set(A),fun(nat,$o),aTP_Lamp_up(fun(nat,set(A)),fun(set(A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),Uu,Uub)),Uua) ) ) ).

% ATP.lambda_529
tff(fact_8709_ATP_Olambda__530,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_530
tff(fact_8710_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_nb(fun(B,set(A)),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua) ) ).

% ATP.lambda_531
tff(fact_8711_ATP_Olambda__532,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_532
tff(fact_8712_ATP_Olambda__533,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uw(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_533
tff(fact_8713_ATP_Olambda__534,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ut(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_534
tff(fact_8714_ATP_Olambda__535,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_un(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_535
tff(fact_8715_ATP_Olambda__536,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_fe(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uu,Uub),Uua) ) ).

% ATP.lambda_536
tff(fact_8716_ATP_Olambda__537,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_kl(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_537
tff(fact_8717_ATP_Olambda__538,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_qy(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_538
tff(fact_8718_ATP_Olambda__539,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_539
tff(fact_8719_ATP_Olambda__540,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_uj(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_540
tff(fact_8720_ATP_Olambda__541,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ug(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_541
tff(fact_8721_ATP_Olambda__542,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,$o),aTP_Lamp_uo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_542
tff(fact_8722_ATP_Olambda__543,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_tw(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_543
tff(fact_8723_ATP_Olambda__544,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_qs(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_544
tff(fact_8724_ATP_Olambda__545,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ef(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_545
tff(fact_8725_ATP_Olambda__546,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),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_546
tff(fact_8726_ATP_Olambda__547,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_qc(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_547
tff(fact_8727_ATP_Olambda__548,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_nd(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_548
tff(fact_8728_ATP_Olambda__549,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_qv(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_549
tff(fact_8729_ATP_Olambda__550,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_nw(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_550
tff(fact_8730_ATP_Olambda__551,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_ts(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_551
tff(fact_8731_ATP_Olambda__552,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_rf(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_552
tff(fact_8732_ATP_Olambda__553,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_tn(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_553
tff(fact_8733_ATP_Olambda__554,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qe(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_554
tff(fact_8734_ATP_Olambda__555,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_zu(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_555
tff(fact_8735_ATP_Olambda__556,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_ms(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_556
tff(fact_8736_ATP_Olambda__557,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_md(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_557
tff(fact_8737_ATP_Olambda__558,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_tz(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_558
tff(fact_8738_ATP_Olambda__559,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_ru(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_559
tff(fact_8739_ATP_Olambda__560,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_oc(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_560
tff(fact_8740_ATP_Olambda__561,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ta(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_561
tff(fact_8741_ATP_Olambda__562,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_xb(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_562
tff(fact_8742_ATP_Olambda__563,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_yc(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_563
tff(fact_8743_ATP_Olambda__564,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_yj(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_564
tff(fact_8744_ATP_Olambda__565,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_yg(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_565
tff(fact_8745_ATP_Olambda__566,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_yk(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_566
tff(fact_8746_ATP_Olambda__567,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_yi(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_567
tff(fact_8747_ATP_Olambda__568,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_yh(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_568
tff(fact_8748_ATP_Olambda__569,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aiw(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> member(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_569
tff(fact_8749_ATP_Olambda__570,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,$o),aTP_Lamp_xq(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(A,aa(B,A,Uua,Uub),Uu) ) ) ).

% ATP.lambda_570
tff(fact_8750_ATP_Olambda__571,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_xw(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu) ) ).

% ATP.lambda_571
tff(fact_8751_ATP_Olambda__572,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ahe(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_572
tff(fact_8752_ATP_Olambda__573,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aja(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_573
tff(fact_8753_ATP_Olambda__574,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bu(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_574
tff(fact_8754_ATP_Olambda__575,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jd(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_575
tff(fact_8755_ATP_Olambda__576,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_ge(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_576
tff(fact_8756_ATP_Olambda__577,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_kd(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_577
tff(fact_8757_ATP_Olambda__578,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(real,fun(A,$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_578
tff(fact_8758_ATP_Olambda__579,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_il(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_579
tff(fact_8759_ATP_Olambda__580,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ym(fun(A,B),fun(B,fun(A,$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_580
tff(fact_8760_ATP_Olambda__581,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,$o),aTP_Lamp_zr(fun(A,A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( ? [X2: A] :
                ( ( Uub = aa(A,A,Uu,X2) )
                & aa(A,$o,Uua,X2) )
            | ? [M8: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X2: A] :
                    ( member(A,X2,M8)
                   => aa(A,$o,Uua,X2) ) ) ) ) ) ).

% ATP.lambda_581
tff(fact_8761_ATP_Olambda__582,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_acg(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) ) )
            | ? [X2: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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),X2),Y4) )
            | ? [X2: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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),X2),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X2)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu,Xs3),Ys4) ) ) ) ) ).

% ATP.lambda_582
tff(fact_8762_ATP_Olambda__583,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_abe(set(A),fun(set(A),fun(set(A),$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_583
tff(fact_8763_ATP_Olambda__584,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),$o),aTP_Lamp_ags(set(product_prod(A,A)),fun(set(A),fun(set(A),$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)) )
        & ! [X2: A] :
            ( member(A,X2,Uua)
           => ? [Xa3: A] :
                ( member(A,Xa3,Uub)
                & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X2),Xa3),Uu) ) ) ) ) ).

% ATP.lambda_584
tff(fact_8764_ATP_Olambda__585,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_jb(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_585
tff(fact_8765_ATP_Olambda__586,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_jc(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_586
tff(fact_8766_ATP_Olambda__587,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_ja(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_587
tff(fact_8767_ATP_Olambda__588,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_yf(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_588
tff(fact_8768_ATP_Olambda__589,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cu(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_589
tff(fact_8769_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(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_590
tff(fact_8770_ATP_Olambda__591,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aaf(A,fun(set(A),fun(A,$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)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_591
tff(fact_8771_ATP_Olambda__592,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_dj(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_592
tff(fact_8772_ATP_Olambda__593,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_593
tff(fact_8773_ATP_Olambda__594,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_ci(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_594
tff(fact_8774_ATP_Olambda__595,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_co(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_595
tff(fact_8775_ATP_Olambda__596,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_aex(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)),product_Pair(A,product_prod(B,C),Uu),aa(C,product_prod(B,C),product_Pair(B,C,Uua),Uub)) ).

% ATP.lambda_596
tff(fact_8776_ATP_Olambda__597,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_aev(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)),product_Pair(A,product_prod(B,C),Uua),aa(C,product_prod(B,C),product_Pair(B,C,Uu),Uub)) ).

% ATP.lambda_597
tff(fact_8777_ATP_Olambda__598,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_ace(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_598
tff(fact_8778_ATP_Olambda__599,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_aga(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_599
tff(fact_8779_ATP_Olambda__600,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_uq(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu,Uua)) ) ).

% ATP.lambda_600
tff(fact_8780_ATP_Olambda__601,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_ahy(fun(A,B),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(B,Uub,aa(set(A),set(B),image2(A,B,Uu),Uua)) ) ) ).

% ATP.lambda_601
tff(fact_8781_ATP_Olambda__602,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_mz(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_602
tff(fact_8782_ATP_Olambda__603,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_my(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_603
tff(fact_8783_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_uu(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_604
tff(fact_8784_ATP_Olambda__605,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uy(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_605
tff(fact_8785_ATP_Olambda__606,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_606
tff(fact_8786_ATP_Olambda__607,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ul(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_607
tff(fact_8787_ATP_Olambda__608,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ui(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_608
tff(fact_8788_ATP_Olambda__609,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ago(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_609
tff(fact_8789_ATP_Olambda__610,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_um(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_610
tff(fact_8790_ATP_Olambda__611,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_611
tff(fact_8791_ATP_Olambda__612,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_tv(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_612
tff(fact_8792_ATP_Olambda__613,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_ej(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_613
tff(fact_8793_ATP_Olambda__614,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ee(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_614
tff(fact_8794_ATP_Olambda__615,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),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_615
tff(fact_8795_ATP_Olambda__616,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_qd(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_616
tff(fact_8796_ATP_Olambda__617,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_qt(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_617
tff(fact_8797_ATP_Olambda__618,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_wb(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_618
tff(fact_8798_ATP_Olambda__619,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_mv(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_619
tff(fact_8799_ATP_Olambda__620,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_620
tff(fact_8800_ATP_Olambda__621,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_zv(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_621
tff(fact_8801_ATP_Olambda__622,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_mt(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_622
tff(fact_8802_ATP_Olambda__623,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_mc(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_623
tff(fact_8803_ATP_Olambda__624,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_ra(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_624
tff(fact_8804_ATP_Olambda__625,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_km(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> dvd_dvd(B,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_625
tff(fact_8805_ATP_Olambda__626,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_xc(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_626
tff(fact_8806_ATP_Olambda__627,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_aeh(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),aa(C,B,Uu,Uub)) ).

% ATP.lambda_627
tff(fact_8807_ATP_Olambda__628,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(B,A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_aif(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = image(B,A,Uu,aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_628
tff(fact_8808_ATP_Olambda__629,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_mp(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_629
tff(fact_8809_ATP_Olambda__630,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(A)),Uub: C] : aa(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_xx(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image2(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_630
tff(fact_8810_ATP_Olambda__631,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_wq(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_631
tff(fact_8811_ATP_Olambda__632,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_agv(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        | aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_632
tff(fact_8812_ATP_Olambda__633,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_aai(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_633
tff(fact_8813_ATP_Olambda__634,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_agg(fun(list(A),A),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_634
tff(fact_8814_ATP_Olambda__635,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_xy(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_635
tff(fact_8815_ATP_Olambda__636,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ey(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_636
tff(fact_8816_ATP_Olambda__637,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_afk(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(list(A),set(A),set2(A),nths(A,Uu,Uua))) ) ).

% ATP.lambda_637
tff(fact_8817_ATP_Olambda__638,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_vf(fun(A,B),fun(real,fun(A,$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_638
tff(fact_8818_ATP_Olambda__639,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_acw(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_639
tff(fact_8819_ATP_Olambda__640,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_eq(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_640
tff(fact_8820_ATP_Olambda__641,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_dc(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_641
tff(fact_8821_ATP_Olambda__642,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_642
tff(fact_8822_ATP_Olambda__643,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_re(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_643
tff(fact_8823_ATP_Olambda__644,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_nr(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_644
tff(fact_8824_ATP_Olambda__645,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_tu(fun(nat,$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_645
tff(fact_8825_ATP_Olambda__646,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_mw(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_646
tff(fact_8826_ATP_Olambda__647,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_et(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_647
tff(fact_8827_ATP_Olambda__648,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_648
tff(fact_8828_ATP_Olambda__649,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_ub(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_649
tff(fact_8829_ATP_Olambda__650,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_ei(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_650
tff(fact_8830_ATP_Olambda__651,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_cg(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_651
tff(fact_8831_ATP_Olambda__652,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_us(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_652
tff(fact_8832_ATP_Olambda__653,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_rg(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_653
tff(fact_8833_ATP_Olambda__654,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_rr(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_654
tff(fact_8834_ATP_Olambda__655,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_nq(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_655
tff(fact_8835_ATP_Olambda__656,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_rj(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_656
tff(fact_8836_ATP_Olambda__657,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_qg(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_657
tff(fact_8837_ATP_Olambda__658,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_em(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_658
tff(fact_8838_ATP_Olambda__659,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_abr(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_659
tff(fact_8839_ATP_Olambda__660,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ne(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_660
tff(fact_8840_ATP_Olambda__661,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_ju(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_661
tff(fact_8841_ATP_Olambda__662,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_ls(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_662
tff(fact_8842_ATP_Olambda__663,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_ahg(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_663
tff(fact_8843_ATP_Olambda__664,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_xi(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_664
tff(fact_8844_ATP_Olambda__665,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_wf(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_665
tff(fact_8845_ATP_Olambda__666,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_wo(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_666
tff(fact_8846_ATP_Olambda__667,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pd(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_667
tff(fact_8847_ATP_Olambda__668,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_sx(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_668
tff(fact_8848_ATP_Olambda__669,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vn(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_669
tff(fact_8849_ATP_Olambda__670,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_ahs(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_670
tff(fact_8850_ATP_Olambda__671,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_xg(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_671
tff(fact_8851_ATP_Olambda__672,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_abp(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_672
tff(fact_8852_ATP_Olambda__673,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_abq(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_673
tff(fact_8853_ATP_Olambda__674,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_ng(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_674
tff(fact_8854_ATP_Olambda__675,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_rd(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_675
tff(fact_8855_ATP_Olambda__676,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_ti(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_676
tff(fact_8856_ATP_Olambda__677,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_ait(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_677
tff(fact_8857_ATP_Olambda__678,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_add(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_678
tff(fact_8858_ATP_Olambda__679,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_adc(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_679
tff(fact_8859_ATP_Olambda__680,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_mk(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_680
tff(fact_8860_ATP_Olambda__681,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xd(fun(B,$o),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_681
tff(fact_8861_ATP_Olambda__682,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),$o)),aTP_Lamp_ahj(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = product_case_prod(A,A,$o,aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_ahi(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ).

% ATP.lambda_682
tff(fact_8862_ATP_Olambda__683,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_qb(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_qa(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_683
tff(fact_8863_ATP_Olambda__684,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_px(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_pw(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_684
tff(fact_8864_ATP_Olambda__685,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_we(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),Uub)),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_685
tff(fact_8865_ATP_Olambda__686,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_wd(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)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_686
tff(fact_8866_ATP_Olambda__687,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_wm(A,fun(set(A),fun(set(A),filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uub),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_687
tff(fact_8867_ATP_Olambda__688,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_fi(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_688
tff(fact_8868_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aar(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_689
tff(fact_8869_ATP_Olambda__690,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aas(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_690
tff(fact_8870_ATP_Olambda__691,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_ws(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_691
tff(fact_8871_ATP_Olambda__692,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_wr(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_692
tff(fact_8872_ATP_Olambda__693,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_kn(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ dvd_dvd(B,Uua,aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_693
tff(fact_8873_ATP_Olambda__694,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: list(B)] : aa(list(B),fun(list(A),list(A)),aa(B,fun(list(B),fun(list(A),list(A))),aTP_Lamp_aie(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),Uu),Uua),Uub) = aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_694
tff(fact_8874_ATP_Olambda__695,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_jl(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bw(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_695
tff(fact_8875_ATP_Olambda__696,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),$o),aTP_Lamp_abw(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [I4: nat] :
          ( ( Uub = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Uu),I4)),aa(nat,B,nth(B,Uua),I4)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_696
tff(fact_8876_ATP_Olambda__697,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_abc(list(A),fun(set(nat),fun(A,$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_697
tff(fact_8877_ATP_Olambda__698,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_abn(set(A),fun(A,fun(A,$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_698
tff(fact_8878_ATP_Olambda__699,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_abl(set(A),fun(A,fun(A,$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_699
tff(fact_8879_ATP_Olambda__700,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_yu(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X2: B] :
          ( ( Uub = aa(B,A,Uu,X2) )
          & member(B,X2,Uua) ) ) ).

% ATP.lambda_700
tff(fact_8880_ATP_Olambda__701,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_yt(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X2: B] :
          ( ( Uub = aa(B,A,Uu,X2) )
          & aa(B,$o,Uua,X2) ) ) ).

% ATP.lambda_701
tff(fact_8881_ATP_Olambda__702,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_yw(fun(A,$o),fun(fun(A,B),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X2: A] :
          ( ( Uub = aa(A,B,Uua,X2) )
          & aa(A,$o,Uu,X2) ) ) ).

% ATP.lambda_702
tff(fact_8882_ATP_Olambda__703,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,$o),aTP_Lamp_zd(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X2: A] :
          ( member(A,X2,Uu)
         => aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X2) ) ) ).

% ATP.lambda_703
tff(fact_8883_ATP_Olambda__704,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,$o),aTP_Lamp_agt(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X2: A] :
          ( member(A,X2,Uu)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X2) ) ) ).

% ATP.lambda_704
tff(fact_8884_ATP_Olambda__705,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aiv(fun(A,B),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X2: A] :
          ( member(A,X2,Uua)
          & ( aa(A,B,Uu,X2) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_705
tff(fact_8885_ATP_Olambda__706,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aha(fun(B,option(A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X2: B] :
          ( member(B,X2,Uua)
          & ( aa(B,option(A),Uu,X2) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_706
tff(fact_8886_ATP_Olambda__707,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,$o),aTP_Lamp_vk(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),N2)
             => 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,N2)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_707
tff(fact_8887_ATP_Olambda__708,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_agu(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X2: B] :
          ( member(B,X2,Uua)
          & ( Uub = aa(B,A,Uu,X2) ) ) ) ).

% ATP.lambda_708
tff(fact_8888_ATP_Olambda__709,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,$o),aTP_Lamp_vy(fun(nat,A),fun(fun(nat,real),fun(nat,$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_709
tff(fact_8889_ATP_Olambda__710,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_yr(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_710
tff(fact_8890_ATP_Olambda__711,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_zj(fun(A,A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ? [N2: 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)),N2),Uu),Uua) ) ).

% ATP.lambda_711
tff(fact_8891_ATP_Olambda__712,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),$o),aTP_Lamp_acm(set(product_prod(A,A)),fun(list(A),fun(list(A),$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),Aa2: A,B7: A,Va4: list(A),W3: list(A)] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Aa2),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),Aa2),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_712
tff(fact_8892_ATP_Olambda__713,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_abo(set(A),fun(set(A),fun(A,$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_713
tff(fact_8893_ATP_Olambda__714,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_abm(set(A),fun(set(A),fun(A,$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_714
tff(fact_8894_ATP_Olambda__715,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_ack(set(A),fun(set(list(A)),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [X2: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
          & member(A,X2,Uu)
          & member(list(A),Xs3,Uua) ) ) ).

% ATP.lambda_715
tff(fact_8895_ATP_Olambda__716,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),$o),aTP_Lamp_acl(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [Us2: list(A),Z6: A,Z9: A,Vs3: 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),Z6),Vs3)) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Z6),Z9),Uu)
          & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z9),Vs3)) ) ) ) ).

% ATP.lambda_716
tff(fact_8896_ATP_Olambda__717,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(
            ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uub)
            & 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_717
tff(fact_8897_ATP_Olambda__718,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_hz(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)),Uub)
            & ~ 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_718
tff(fact_8898_ATP_Olambda__719,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_id(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(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_719
tff(fact_8899_ATP_Olambda__720,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_aac(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_720
tff(fact_8900_ATP_Olambda__721,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_aid(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uua,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),Uuc),Uuc) ).

% ATP.lambda_721
tff(fact_8901_ATP_Olambda__722,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_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,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_722
tff(fact_8902_ATP_Olambda__723,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_gn(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_723
tff(fact_8903_ATP_Olambda__724,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_wc(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_724
tff(fact_8904_ATP_Olambda__725,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_go(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_725
tff(fact_8905_ATP_Olambda__726,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_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_726
tff(fact_8906_ATP_Olambda__727,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_zz(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_727
tff(fact_8907_ATP_Olambda__728,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_ev(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_728
tff(fact_8908_ATP_Olambda__729,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_ew(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_729
tff(fact_8909_ATP_Olambda__730,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_cc(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_730
tff(fact_8910_ATP_Olambda__731,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_jq(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_731
tff(fact_8911_ATP_Olambda__732,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_aiu(set(A),fun(B,fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uu),Uua,Uub) ).

% ATP.lambda_732
tff(fact_8912_ATP_Olambda__733,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_agy(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)),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)),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_733
tff(fact_8913_ATP_Olambda__734,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
      aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_zs(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_734
tff(fact_8914_ATP_Olambda__735,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_kk(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_735
tff(fact_8915_ATP_Olambda__736,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_kj(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_736
tff(fact_8916_ATP_Olambda__737,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_abb(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)),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_aba(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_737
tff(fact_8917_ATP_Olambda__738,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_aav(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)),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_aau(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_738
tff(fact_8918_ATP_Olambda__739,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_ko(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_739
tff(fact_8919_ATP_Olambda__740,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(list(C),fun(B,B))),Uua: fun(A,C),Uub: A,Uuc: list(A)] : aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),aa(fun(A,C),fun(A,fun(list(A),fun(B,B))),aTP_Lamp_aii(fun(C,fun(list(C),fun(B,B))),fun(fun(A,C),fun(A,fun(list(A),fun(B,B)))),Uu),Uua),Uub),Uuc) = aa(list(C),fun(B,B),aa(C,fun(list(C),fun(B,B)),Uu,aa(A,C,Uua,Uub)),aa(list(A),list(C),map(A,C,Uua),Uuc)) ).

% ATP.lambda_740
tff(fact_8920_ATP_Olambda__741,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,$o)),aTP_Lamp_agi(fun(A,fun(A,$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_741
tff(fact_8921_ATP_Olambda__742,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_aej(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_742
tff(fact_8922_ATP_Olambda__743,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_zn(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_743
tff(fact_8923_ATP_Olambda__744,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_cy(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_cx(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_744
tff(fact_8924_ATP_Olambda__745,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_gp(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_745
tff(fact_8925_ATP_Olambda__746,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_ec(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_eb(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_746
tff(fact_8926_ATP_Olambda__747,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_bz(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aTP_Lamp_by(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_an(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_747
tff(fact_8927_ATP_Olambda__748,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_gk(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_gj(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_748
tff(fact_8928_ATP_Olambda__749,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_gh(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_gg(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_749
tff(fact_8929_ATP_Olambda__750,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_op(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_750
tff(fact_8930_ATP_Olambda__751,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_on(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_751
tff(fact_8931_ATP_Olambda__752,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_ol(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_752
tff(fact_8932_ATP_Olambda__753,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_ok(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_753
tff(fact_8933_ATP_Olambda__754,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_df(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_754
tff(fact_8934_ATP_Olambda__755,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_kq(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).

% ATP.lambda_755
tff(fact_8935_ATP_Olambda__756,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_cx(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_756
tff(fact_8936_ATP_Olambda__757,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_757
tff(fact_8937_ATP_Olambda__758,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_nm(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Uub),Uuc),Uua) ) ) ) ).

% ATP.lambda_758
tff(fact_8938_ATP_Olambda__759,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_uc(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_759
tff(fact_8939_ATP_Olambda__760,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_vc(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_760
tff(fact_8940_ATP_Olambda__761,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_hc(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_761
tff(fact_8941_ATP_Olambda__762,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_ha(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_762
tff(fact_8942_ATP_Olambda__763,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_hb(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_763
tff(fact_8943_ATP_Olambda__764,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_ii(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_764
tff(fact_8944_ATP_Olambda__765,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_acp(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),X2: A,Y4: A,Xs5: list(A),Ys6: 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),X2),Xs5)) )
            & ( 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),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X2),Y4),Uu) ) ) ) ).

% ATP.lambda_765
tff(fact_8945_ATP_Olambda__766,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_dh(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_766
tff(fact_8946_ATP_Olambda__767,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_lq(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,aa(set(B),set(A),image2(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_767
tff(fact_8947_ATP_Olambda__768,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_bw(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_768
tff(fact_8948_ATP_Olambda__769,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_an(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_769
tff(fact_8949_ATP_Olambda__770,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_dg(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_770
tff(fact_8950_ATP_Olambda__771,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_vr(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_771
tff(fact_8951_ATP_Olambda__772,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_lp(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_772
tff(fact_8952_ATP_Olambda__773,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_lw(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_773
tff(fact_8953_ATP_Olambda__774,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_ma(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_774
tff(fact_8954_ATP_Olambda__775,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_ajb(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_775
tff(fact_8955_ATP_Olambda__776,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_xv(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_776
tff(fact_8956_ATP_Olambda__777,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_di(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_777
tff(fact_8957_ATP_Olambda__778,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_gz(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_778
tff(fact_8958_ATP_Olambda__779,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_kw(nat,fun(nat,fun(nat,fun(nat,$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_779
tff(fact_8959_ATP_Olambda__780,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ajd(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_780
tff(fact_8960_ATP_Olambda__781,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_ku(nat,fun(nat,fun(nat,fun(nat,$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_781
tff(fact_8961_ATP_Olambda__782,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),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_782
tff(fact_8962_ATP_Olambda__783,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_la(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).

% ATP.lambda_783
tff(fact_8963_ATP_Olambda__784,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_act(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),product_Pair(A,B,Uu),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_784
tff(fact_8964_ATP_Olambda__785,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_acs(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),product_Pair(A,B,Uub),Uu)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_785
tff(fact_8965_ATP_Olambda__786,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_lr(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_786
tff(fact_8966_ATP_Olambda__787,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_ay(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_787
tff(fact_8967_ATP_Olambda__788,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_ba(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_788
tff(fact_8968_ATP_Olambda__789,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_sv(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_789
tff(fact_8969_ATP_Olambda__790,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_fo(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_790
tff(fact_8970_ATP_Olambda__791,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_jg(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_791
tff(fact_8971_ATP_Olambda__792,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_pw(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_792
tff(fact_8972_ATP_Olambda__793,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_qa(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_793
tff(fact_8973_ATP_Olambda__794,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_ov(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_794
tff(fact_8974_ATP_Olambda__795,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_fc(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_795
tff(fact_8975_ATP_Olambda__796,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_ve(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)),insert2(B),Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_796
tff(fact_8976_ATP_Olambda__797,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,$o)),aTP_Lamp_ahk(fun(A,$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_797
tff(fact_8977_ATP_Olambda__798,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_agb(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_798
tff(fact_8978_ATP_Olambda__799,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_gj(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_799
tff(fact_8979_ATP_Olambda__800,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_gg(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_800
tff(fact_8980_ATP_Olambda__801,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_wz(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_801
tff(fact_8981_ATP_Olambda__802,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_aef(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),product_Pair(A,B,aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_802
tff(fact_8982_ATP_Olambda__803,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_agk(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_803
tff(fact_8983_ATP_Olambda__804,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_ea(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uuc)),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bw(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_804
tff(fact_8984_ATP_Olambda__805,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_bx(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uuc)),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bw(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_805
tff(fact_8985_ATP_Olambda__806,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_pl(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_806
tff(fact_8986_ATP_Olambda__807,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_ot(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_807
tff(fact_8987_ATP_Olambda__808,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_pn(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_808
tff(fact_8988_ATP_Olambda__809,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_sw(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_809
tff(fact_8989_ATP_Olambda__810,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,$o)),aTP_Lamp_ajh(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
       => aa(A,$o,aa(A,fun(A,$o),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ) ).

% ATP.lambda_810
tff(fact_8990_ATP_Olambda__811,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_abh(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_811
tff(fact_8991_ATP_Olambda__812,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_vd(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_812
tff(fact_8992_ATP_Olambda__813,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_tf(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_te(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_te(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_te(A,A)))))) ) ).

% ATP.lambda_813
tff(fact_8993_ATP_Olambda__814,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_tc(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_814
tff(fact_8994_ATP_Olambda__815,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_tb(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_815
tff(fact_8995_ATP_Olambda__816,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ma(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_816
tff(fact_8996_ATP_Olambda__817,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),$o)),aTP_Lamp_acf(fun(A,fun(A,$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) ) )
        | ? [X2: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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,X2),Y4) )
        | ? [X2: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),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,X2),Y4)
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,Y4),X2)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uua,Xs3),Ys4) ) ) ) ).

% ATP.lambda_817
tff(fact_8997_ATP_Olambda__818,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_acc(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_818
tff(fact_8998_ATP_Olambda__819,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_aiz(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),vimage(C,A,Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_819
tff(fact_8999_ATP_Olambda__820,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ly(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Uua),collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_lw(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_820
tff(fact_9000_ATP_Olambda__821,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_me(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),Uub),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ma(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_821
tff(fact_9001_ATP_Olambda__822,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_lx(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Uua),collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_lw(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_822
tff(fact_9002_ATP_Olambda__823,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_mb(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),Uub),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ma(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_823
tff(fact_9003_ATP_Olambda__824,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_tt(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_fh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uua)))) ) ) ).

% ATP.lambda_824
tff(fact_9004_ATP_Olambda__825,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_acy(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_825
tff(fact_9005_ATP_Olambda__826,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_acu(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_826
tff(fact_9006_ATP_Olambda__827,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_iy(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_827
tff(fact_9007_ATP_Olambda__828,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_iw(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_828
tff(fact_9008_ATP_Olambda__829,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_er(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_829
tff(fact_9009_ATP_Olambda__830,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_cj(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_830
tff(fact_9010_ATP_Olambda__831,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_aeg(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),product_Pair(B,C,aa(D,B,Uua,Uub)),Uuc)) ).

% ATP.lambda_831
tff(fact_9011_ATP_Olambda__832,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_aee(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),product_Pair(B,C,Uub),aa(D,C,Uua,Uuc))) ).

% ATP.lambda_832
tff(fact_9012_ATP_Olambda__833,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_xa(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_wz(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_833
tff(fact_9013_ATP_Olambda__834,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_agp(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),takeWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),Uu),Uua)),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Uub),Uuc))) ).

% ATP.lambda_834
tff(fact_9014_ATP_Olambda__835,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_pi(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_835
tff(fact_9015_ATP_Olambda__836,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_acq(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),product_Pair(list(A),product_prod(A,list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Uub),Uuc))) ).

% ATP.lambda_836
tff(fact_9016_ATP_Olambda__837,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_aik(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uua,Uuc)),image(B,B,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),bot_bot(set(B)))))) ).

% ATP.lambda_837
tff(fact_9017_ATP_Olambda__838,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_vu(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X2: A] :
          ( ( member(A,X2,Uu)
           => member(B,aa(A,B,Uuc,X2),Uua) )
          & ( ~ member(A,X2,Uu)
           => ( aa(A,B,Uuc,X2) = Uub ) ) ) ) ).

% ATP.lambda_838
tff(fact_9018_ATP_Olambda__839,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,$o)),aTP_Lamp_yv(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [X2: A,Y4: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X2),Y4) )
          & aa(A,$o,Uu,X2)
          & aa(B,$o,Uua,Y4) ) ) ).

% ATP.lambda_839
tff(fact_9019_ATP_Olambda__840,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_aau(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert2(product_prod(C,B)),aa(B,product_prod(C,B),product_Pair(C,B,Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_840
tff(fact_9020_ATP_Olambda__841,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_aba(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(product_prod(A,C),fun(set(product_prod(A,C)),set(product_prod(A,C))),insert2(product_prod(A,C)),aa(C,product_prod(A,C),product_Pair(A,C,Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_841
tff(fact_9021_ATP_Olambda__842,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,fun(A,B)),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: C] : aa(C,B,aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_pe(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_842
tff(fact_9022_ATP_Olambda__843,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_pt(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_ps(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_843
tff(fact_9023_ATP_Olambda__844,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_gr(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_gq(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_844
tff(fact_9024_ATP_Olambda__845,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_oo(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_on(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_845
tff(fact_9025_ATP_Olambda__846,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_fd(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_fc(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_846
tff(fact_9026_ATP_Olambda__847,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hr(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_847
tff(fact_9027_ATP_Olambda__848,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_hn(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_848
tff(fact_9028_ATP_Olambda__849,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_ho(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_849
tff(fact_9029_ATP_Olambda__850,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,$o))),aTP_Lamp_ahi(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uud),bot_bot(set(A))))))),field2(A,Uu))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),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),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),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_850
tff(fact_9030_ATP_Olambda__851,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_gq(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_851
tff(fact_9031_ATP_Olambda__852,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_yd(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_852
tff(fact_9032_ATP_Olambda__853,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_ps(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_pq(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)),insert2(A),Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_853
tff(fact_9033_ATP_Olambda__854,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_pg(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_854
tff(fact_9034_ATP_Olambda__855,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_pv(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_855
tff(fact_9035_ATP_Olambda__856,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_ox(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_856
tff(fact_9036_ATP_Olambda__857,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_pc(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_857
tff(fact_9037_ATP_Olambda__858,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_pp(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_858
tff(fact_9038_ATP_Olambda__859,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_yq(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) ) )
        & $ite(
            Uue = Uuf,
            ! [X2: vEBT_VEBT] :
              ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X8) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),X2)
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_859
tff(fact_9039_ATP_Olambda__860,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_860
tff(fact_9040_ATP_Olambda__861,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_nc($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_861
tff(fact_9041_ATP_Olambda__862,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_ajc(set(B),fun(A,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_862
tff(fact_9042_ATP_Olambda__863,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_adi(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_863
tff(fact_9043_ATP_Olambda__864,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_adh(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_864
tff(fact_9044_ATP_Olambda__865,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_mn(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_865
tff(fact_9045_ATP_Olambda__866,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_adl(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_866
tff(fact_9046_ATP_Olambda__867,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,fun(B,$o),aTP_Lamp_xe(fun(B,$o),fun(A,fun(B,$o)),Uu),Uua) = Uu ).

% ATP.lambda_867
tff(fact_9047_ATP_Olambda__868,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_aaq(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_868
tff(fact_9048_ATP_Olambda__869,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_lj(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_869
tff(fact_9049_ATP_Olambda__870,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_lk(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_870
tff(fact_9050_ATP_Olambda__871,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_oy(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_871
tff(fact_9051_ATP_Olambda__872,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ll(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_872
tff(fact_9052_ATP_Olambda__873,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_rc(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_873
tff(fact_9053_ATP_Olambda__874,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_qf(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_874
tff(fact_9054_ATP_Olambda__875,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_lv(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_875
tff(fact_9055_ATP_Olambda__876,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_mm(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_876
tff(fact_9056_ATP_Olambda__877,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ff(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_877
tff(fact_9057_ATP_Olambda__878,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_np(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_878
tff(fact_9058_ATP_Olambda__879,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_lz(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_879
tff(fact_9059_ATP_Olambda__880,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_ahf(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_880
tff(fact_9060_ATP_Olambda__881,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_afq(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_881
tff(fact_9061_ATP_Olambda__882,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_qh(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_882
tff(fact_9062_ATP_Olambda__883,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_adm(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_883
tff(fact_9063_ATP_Olambda__884,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aek(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_884
tff(fact_9064_ATP_Olambda__885,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_lu(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_885
tff(fact_9065_ATP_Olambda__886,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_acv(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_886
tff(fact_9066_ATP_Olambda__887,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_ada(A,fun(list(A),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_887
tff(fact_9067_ATP_Olambda__888,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_acz(A,fun(list(A),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_888
tff(fact_9068_ATP_Olambda__889,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agj(A,fun(A,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_889
tff(fact_9069_ATP_Olambda__890,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_cr(complex,complex),Uu) = Uu ).

% ATP.lambda_890
tff(fact_9070_ATP_Olambda__891,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_bk(nat,nat),Uu) = Uu ).

% ATP.lambda_891
tff(fact_9071_ATP_Olambda__892,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_bj(int,int),Uu) = Uu ).

% ATP.lambda_892
tff(fact_9072_ATP_Olambda__893,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_te(A,A),Uu) = Uu ) ).

% ATP.lambda_893
tff(fact_9073_ATP_Olambda__894,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_th(A,A),Uu) = Uu ) ).

% ATP.lambda_894
tff(fact_9074_ATP_Olambda__895,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_tj(A,A),Uu) = Uu ) ).

% ATP.lambda_895
tff(fact_9075_ATP_Olambda__896,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_afl(A,A),Uu) = Uu ) ).

% ATP.lambda_896
tff(fact_9076_ATP_Olambda__897,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_xr(A,A),Uu) = Uu ) ).

% ATP.lambda_897
tff(fact_9077_ATP_Olambda__898,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ap(A,A),Uu) = Uu ) ).

% ATP.lambda_898
tff(fact_9078_ATP_Olambda__899,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_lg(A,A),Uu) = Uu ).

% ATP.lambda_899
tff(fact_9079_ATP_Olambda__900,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_xm(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_900
tff(fact_9080_ATP_Olambda__901,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_mr(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_901
tff(fact_9081_ATP_Olambda__902,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_lh(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_902
tff(fact_9082_ATP_Olambda__903,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_li(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_903
tff(fact_9083_ATP_Olambda__904,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_xn(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_904
tff(fact_9084_ATP_Olambda__905,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_adj(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_905
tff(fact_9085_ATP_Olambda__906,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_dt(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_906
tff(fact_9086_ATP_Olambda__907,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_dn(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_907
tff(fact_9087_ATP_Olambda__908,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_bl(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_908
tff(fact_9088_ATP_Olambda__909,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_afg(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_909
tff(fact_9089_ATP_Olambda__910,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_oz(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_910
tff(fact_9090_ATP_Olambda__911,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ao(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_911
tff(fact_9091_ATP_Olambda__912,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_abz(B,option(A)),Uu) = none(A) ).

% ATP.lambda_912
tff(fact_9092_ATP_Olambda__913,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_aca(A,option(B)),Uu) = none(B) ).

% ATP.lambda_913
tff(fact_9093_ATP_Olambda__914,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_ads(A,B),Uu) = undefined(B) ).

% ATP.lambda_914
tff(fact_9094_ATP_Olambda__915,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_jv(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_915
tff(fact_9095_ATP_Olambda__916,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_am(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_916
tff(fact_9096_ATP_Olambda__917,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_jw(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_917
tff(fact_9097_ATP_Olambda__918,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_nf(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_918
tff(fact_9098_ATP_Olambda__919,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_aal(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_919

% Type constructors (827)
tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple592849572758109894attice(A18)
     => counta4013691401010221786attice(fun(A17,A18)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => condit1219197933456340205attice(fun(A17,A18)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( counta3822494911875563373attice(A18)
     => counta3822494911875563373attice(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple592849572758109894attice(A18)
     => comple592849572758109894attice(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4967611905675639751up_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4346867609351753570nf_top(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple6319245703460814977attice(fun(A17,A18)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A17: $tType,A18: $tType] :
      ( boolea8198339166811842893lgebra(A18)
     => boolea8198339166811842893lgebra(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple9053668089753744459l_ccpo(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A17: $tType,A18: $tType] :
      ( semilattice_sup(A18)
     => semilattice_sup(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A17: $tType,A18: $tType] :
      ( semilattice_inf(A18)
     => semilattice_inf(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( distrib_lattice(A18)
     => distrib_lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order_top(A18)
     => order_top(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order_bot(A18)
     => order_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Countable_Ocountable,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & countable(A18) )
     => countable(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A17: $tType,A18: $tType] :
      ( preorder(A18)
     => preorder(fun(A17,A18)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( lattice(A18)
     => lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order(A18)
     => order(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A17: $tType,A18: $tType] :
      ( top(A18)
     => top(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ord(A18)
     => ord(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bot(A18)
     => bot(fun(A17,A18)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A17: $tType,A18: $tType] :
      ( uminus(A18)
     => uminus(fun(A17,A18)) ) ).

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___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_11,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_12,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_13,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_14,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_15,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_16,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_17,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_18,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_19,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_20,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_21,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_22,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_23,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_24,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_25,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_26,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_27,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_28,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_29,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_30,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_31,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_32,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_33,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_34,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_35,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_36,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_37,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_38,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_39,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_40,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_41,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_42,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_43,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_44,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_45,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_47,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_48,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_51,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_52,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_53,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_54,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_55,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_56,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60,axiom,
    comm_monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
    comm_monoid_diff(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__add_61,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_62,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_63,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_65,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_68,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_69,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_70,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_71,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_72,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_73,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_74,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_75,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_76,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Countable_Ocountable_77,axiom,
    countable(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_78,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_79,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_80,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_81,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_82,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_83,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_84,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_85,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_86,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_87,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_88,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_89,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_90,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_91,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_92,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_93,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_94,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Power_Opower_95,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_96,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_97,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_98,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_99,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_100,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_101,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_102,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_103,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_104,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_105,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_106,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_107,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_108,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_109,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_110,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_111,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_112,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_113,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_114,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_115,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_116,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_117,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_118,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_119,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_120,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_121,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_122,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_123,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_124,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_125,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_126,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_127,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_128,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_129,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_130,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_131,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_132,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_133,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_134,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_135,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_136,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_137,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_138,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_139,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_140,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_141,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_142,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_143,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_144,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_145,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_146,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_147,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_148,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_149,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_150,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_151,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_152,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_153,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_154,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_155,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_156,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_157,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_158,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_159,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Countable_Ocountable_160,axiom,
    countable(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_161,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_162,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_163,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_164,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_165,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_166,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_167,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_168,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_169,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_170,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_171,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_172,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_173,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_174,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_175,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_176,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_177,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_178,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_179,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_180,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_181,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_182,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_183,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_184,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_185,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_186,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_187,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_188,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_189,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_190,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_191,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_192,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_193,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_194,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_195,axiom,
    ! [A17: $tType] : counta4013691401010221786attice(set(A17)) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_196,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(set(A17)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_197,axiom,
    ! [A17: $tType] : counta3822494911875563373attice(set(A17)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_198,axiom,
    ! [A17: $tType] : comple592849572758109894attice(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_199,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_200,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(set(A17)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_201,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(set(A17)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_202,axiom,
    ! [A17: $tType] : boolea8198339166811842893lgebra(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_203,axiom,
    ! [A17: $tType] : bounded_lattice_bot(set(A17)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_204,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_205,axiom,
    ! [A17: $tType] : semilattice_sup(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_206,axiom,
    ! [A17: $tType] : semilattice_inf(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_207,axiom,
    ! [A17: $tType] : distrib_lattice(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_208,axiom,
    ! [A17: $tType] : bounded_lattice(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_209,axiom,
    ! [A17: $tType] : order_top(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_210,axiom,
    ! [A17: $tType] : order_bot(set(A17)) ).

tff(tcon_Set_Oset___Countable_Ocountable_211,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => countable(set(A17)) ) ).

tff(tcon_Set_Oset___Orderings_Opreorder_212,axiom,
    ! [A17: $tType] : preorder(set(A17)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_213,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => finite_finite(set(A17)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_214,axiom,
    ! [A17: $tType] : lattice(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder_215,axiom,
    ! [A17: $tType] : order(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Otop_216,axiom,
    ! [A17: $tType] : top(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oord_217,axiom,
    ! [A17: $tType] : ord(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Obot_218,axiom,
    ! [A17: $tType] : bot(set(A17)) ).

tff(tcon_Set_Oset___Groups_Ouminus_219,axiom,
    ! [A17: $tType] : uminus(set(A17)) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_220,axiom,
    counta4013691401010221786attice($o) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_221,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_222,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_223,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_224,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_225,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_226,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_227,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_228,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_229,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_230,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_231,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_232,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_233,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_234,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_235,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_236,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_237,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_238,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_239,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_240,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_241,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Countable_Ocountable_242,axiom,
    countable($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_243,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_244,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_245,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_246,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_247,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_248,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_249,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_250,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_251,axiom,
    uminus($o) ).

tff(tcon_List_Olist___Countable_Ocountable_252,axiom,
    ! [A17: $tType] :
      ( countable(A17)
     => countable(list(A17)) ) ).

tff(tcon_List_Olist___Nat_Osize_253,axiom,
    ! [A17: $tType] : size(list(A17)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_254,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_255,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_256,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_257,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_258,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_259,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_260,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_261,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_262,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_263,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_264,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_265,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_266,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_267,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_268,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_269,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_270,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_271,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_272,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_273,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_274,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_275,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_276,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_277,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_278,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_279,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_280,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_281,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_282,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_283,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_284,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_285,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_286,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_287,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_288,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_289,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_290,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_291,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_292,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_293,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_294,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_295,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_296,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_297,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_298,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_299,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_300,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_301,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_302,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_303,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_304,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_305,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_306,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_307,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_308,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_309,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_310,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_311,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_312,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_313,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_314,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_315,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_316,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_317,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_318,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_319,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_320,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_321,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_322,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_323,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_324,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_325,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_326,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_327,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_328,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_329,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_330,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_331,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_332,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_333,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_334,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_335,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_336,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_337,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_338,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_339,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_340,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_341,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_342,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_343,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_344,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_345,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_346,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_347,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_348,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_349,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_350,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_351,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_352,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_353,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_354,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_355,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_356,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_357,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_358,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_359,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_360,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_361,axiom,
    dvd(real) ).

tff(tcon_Sum__Type_Osum___Countable_Ocountable_362,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( countable(A17)
        & countable(A18) )
     => countable(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_363,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_364,axiom,
    ! [A17: $tType,A18: $tType] : size(sum_sum(A17,A18)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_365,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_366,axiom,
    ! [A17: $tType] : counta3822494911875563373attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_367,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_368,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_369,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_370,axiom,
    ! [A17: $tType] : bounded_lattice_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_371,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_372,axiom,
    ! [A17: $tType] : semilattice_sup(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_373,axiom,
    ! [A17: $tType] : semilattice_inf(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_374,axiom,
    ! [A17: $tType] : distrib_lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_375,axiom,
    ! [A17: $tType] : bounded_lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_376,axiom,
    ! [A17: $tType] : order_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_377,axiom,
    ! [A17: $tType] : order_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_378,axiom,
    ! [A17: $tType] : preorder(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_379,axiom,
    ! [A17: $tType] : lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_380,axiom,
    ! [A17: $tType] : order(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_381,axiom,
    ! [A17: $tType] : top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_382,axiom,
    ! [A17: $tType] : ord(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_383,axiom,
    ! [A17: $tType] : bot(filter(A17)) ).

tff(tcon_Option_Ooption___Countable_Ocountable_384,axiom,
    ! [A17: $tType] :
      ( countable(A17)
     => countable(option(A17)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_385,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => finite_finite(option(A17)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_386,axiom,
    ! [A17: $tType] : size(option(A17)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_387,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_388,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_389,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_390,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_391,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_392,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_393,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_394,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_395,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_396,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_397,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_398,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_399,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_400,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_401,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_402,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_403,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_404,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_405,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_406,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_407,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_408,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_409,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_410,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_411,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_412,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_413,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_414,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_415,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_416,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_417,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_418,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_419,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_420,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_421,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_422,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_423,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_424,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_425,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_426,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_427,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_428,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_429,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_430,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_431,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_432,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_433,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_434,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_435,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_436,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_437,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_438,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_439,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_440,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_441,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_442,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_443,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_444,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_445,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_446,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_447,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_448,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_449,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_450,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_451,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_452,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_453,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_454,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_455,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_456,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_457,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_458,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_459,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_460,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_461,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_462,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_463,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_464,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_465,axiom,
    counta4013691401010221786attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_466,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_467,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_468,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_469,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_470,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_471,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_472,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_473,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_474,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_475,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_476,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_477,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_478,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_479,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_480,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_481,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_482,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_483,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_484,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_485,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_486,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_487,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_488,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_489,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_490,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_491,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_492,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_493,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_494,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_495,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_496,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_497,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_498,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_499,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_500,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_501,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_502,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable_Ocountable_503,axiom,
    countable(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_504,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_505,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_506,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_507,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_508,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_509,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_510,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_511,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_512,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_513,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_514,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_515,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_516,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_517,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_518,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_519,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_520,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_521,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_522,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_523,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_524,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( topolo4958980785337419405_space(A17)
        & topolo4958980785337419405_space(A18) )
     => topolo4958980785337419405_space(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_525,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( topological_t2_space(A17)
        & topological_t2_space(A18) )
     => topological_t2_space(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_526,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( topological_t1_space(A17)
        & topological_t1_space(A18) )
     => topological_t1_space(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Countable_Ocountable_527,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( countable(A17)
        & countable(A18) )
     => countable(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_528,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_529,axiom,
    ! [A17: $tType,A18: $tType] : size(product_prod(A17,A18)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_530,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_531,axiom,
    counta4013691401010221786attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_532,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_533,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_534,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_535,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_536,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_537,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_538,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_539,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_540,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_541,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_542,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_543,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_544,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_545,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_546,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_547,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_548,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable_Ocountable_549,axiom,
    countable(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_550,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_551,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_552,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_553,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_554,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_555,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_556,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_557,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_558,axiom,
    uminus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_559,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_560,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_561,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_562,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_563,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_564,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_565,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_566,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_567,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_568,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_569,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_570,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_571,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_572,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_573,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_574,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_575,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_576,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_577,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_578,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_579,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_580,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_581,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_582,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_583,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_584,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_585,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_586,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_587,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_588,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_589,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_590,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_591,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_592,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_593,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_594,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_595,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_596,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_597,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_598,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_599,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_600,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_601,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_602,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_603,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_604,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_605,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_606,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_607,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_608,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_609,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_610,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_611,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_612,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_613,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_614,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_615,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_616,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_617,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_618,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_619,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_620,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_621,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_622,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_623,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_624,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_625,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_626,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_627,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_628,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_629,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_630,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_631,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_632,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_633,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_634,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_635,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_636,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_637,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_638,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_639,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_640,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_641,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_642,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_643,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_644,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_645,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_646,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_647,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_648,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_649,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_650,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_651,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_652,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_653,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_654,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_655,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_656,axiom,
    size(vEBT_VEBT) ).

% Helper facts (5)
tff(help_fNot_2_1_U,axiom,
    ! [P: $o] :
      ( (P)
      | aa($o,$o,fNot,(P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: $o] :
      ( ~ aa($o,$o,fNot,(P))
      | ~ (P) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ( X9 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X9),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X8: A] : aa(A,$o,P,X8) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ? [Y2: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(mi,na))),Y2) ).

%------------------------------------------------------------------------------